Dynamic Competition with Irreversible Moves

Friedel Bolle and∗Yves Breitmoser†

European University Viadrina

Postfach 1786, 15207 Frankfurt(Oder), Germany

email: yves@euv-frankfurt-o.de

October 12, 2004

Discussion Paper 207

Europa–Universität Viadrina, Frankfurt (Oder)

Abstract

We define a class of duopolistic “competitive games” (including Bertrand and

Cournot competition, public–good games, and rent–seeking contests) and apply it to

define a dynamic game where deviations to increasingly competitive strategies are

irreversible. There is generally a unique profile of equilibrium payoffs that Pareto

dominates the stage game Nash payoffs, and in a wide range of circumstances, this

payoff profile is even unique overall. Moreover, we define a generalized repeated

game where deviations to increasingly competitive strategies can be made irreversible

by the respective player. Under comparably mild assumptions, we find that the sets of

equilibria are payoff–equivalent in both kinds of games. Thus, it is irrelevant whether

deviations to competitive strategies are irreversible or can be made irreversible. Fi-

nally, we define a perturbed infinitely repeated game, where opportunities to restricts

one’s strategies occur with arbitrarily small probabilities, and extend the above equiv-

alence (without strengthening the assumptions).

JEL classification:D40, L10 (Market Structure), D43, L13 (Oligopoly), C73 (Dy-

namic Games)

Keywords: dynamic oligopoly, tacit collusion, equilibrium refinement, irreversible

moves

∗We would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the support.†corresponding author

1 Introduction

“To burn down bridges passed” is a rather frequently cited metaphor describing an irreversible de-

cision. As a result of this decision, another decision (the previous troops movement) ismadeirre-

versible. This is quite remarkable, since at first glance it seems preferable that decisions be reversible.

However, decision makers can be better off deliberately restricting their options; not only to win bat-

tles, but also to deter entrances of firms into markets (through committing to competitive prices,

capacity levels, or advertisement spendings, e.g. Sylos–Labini, 1962), and to cope with the own

weaknesses of will (Strotz, 1956, Elster, 1984, and many others). Moreover, players can be restricted

unvoluntarily through their emotions (e.g. Elster, 1998, and Loewenstein, 2000) or through persons

that the players have to report to (e.g. share holders). We shall analyze these aspects in a generalized

model of duopolistic competition. Precisely, we analyze which outcomes would result

1. when deviations to more competitive strategies are irreversible,

2. when the players are provided with opportunities to make such deviations irreversible, and

3. when the players are provided with such opportunities arbitrarily rarely.

Below, we define precisely when a strategy would be said to be more competitive than another one,

but basically: the more competitive a strategy the less the opponent’s payoff, the own payoff is

unimodal in the competitiveness of the own strategy, and the best reply functions in terms of com-

petitiveness (i.e. mappings of opponent’s competitiveness to optimal own competitiveness) have a

unique intersection. This intersection is unique even after linearly intrapolating the best–reply func-

tions (note that we assume the strategy sets to be finite) and the unique intersection is in a pure

strategy profile. Simultaneous move games where the strategies can be ranked in this sense of com-

petitiveness are calledcompetitive games.

The resulting set of competitive games includes numerous models that have been studied in in-

dustrial organization (Bertrand/Cournot competition, rent–seeking contests, and public–good games),

behavioral economics, and political economics (e.g. arms races, trade tariffs, and environmental reg-

ulations). Interactions that are not competitive in our sense include coordination games and zero–sum

games. Let us note that competitive games in our sense are not related to competitive games as they

are defined in Friedman (1983) and Kats and Thisse (1992). Their classes of games are to generalize

zero–sum games, instead of generalizing duopolistic models of competition (as our definition).

The implications of deviations being strictly irreversible (point 1) are analyzed inirreversible

move games(IMGs): in each stage the players can repeat the previously played stage strategy or

they can deviate to a more competitive one; the game ends when both players repeat their previous

strategies, and only the finally played strategies are payoff–relevant. These games are related to

monotone games (Gale, 2001; Lockwood and Thomas, 2002), which is discussed below, and to

spectrum auctions (Milgrom, 2000; Bolle and Breitmoser, 2001).

1

For a wide range of competitive constituent games, the payoff allocations that are equilibrial

in IMGs are unique under renegotiation proofness (including all games with strategic complements

Bulow et al., 1985). We assume renegotiation proofness in the sense ofPareto perfectness(Bernheim

et al., 1987): the strategy profiles are required to induce Pareto efficient Nash equilibria in all sub-

games. Thus, we rule out threats of inefficient responses to deviations. Moreover, there is generally

a unique equilibrium where no player is worse off than in the Nash equilibrium of the constituent

game, which might be focal out of the players’ eyes (however, we do not subject this claim to more

powerful concepts of equilibrium selection).

Basically, IMGs describe the dynamics in games where the decisions have sunk–cost character.

These include games of uncovering information, the destruction of (unique) objects, and the commit-

ment of crimes (see also Schelling, 1963). Moreover, IMGs describe the dynamics that would apply

in infinite interactions, even though IMGs themselves are finite (but importantly, the time horizon is

endogenous, i.e. indefinite). For instance, the IMG equilibria are payoff–equivalent to the equilibria

of infinite games where all rounds are payoff relevant when the players do not discount future payoffs

(provided the irreversibilities apply as described above). This is the most significant characteristic

that distinguishes our approach to IMGs from irreversible move games with exogenous and finite

horizons (see Romano and Yildirim, 2004). In the latter model, the players can backward induce that

their opponent would deviate to competitive strategies in the last round, and as a result, collusive out-

comes can not be sustained in equilibrium. Moreover, since there is a last round in finite IMGs, the

respective set of equilibrium payoffs does not generalize to infinite games (similarly to the distinction

of finitely and infinitely repeated Prisoner’s dilemmas). This point is discussed below.

Another infinite game that is closely related to IMGs (as we define them) is the scope of point 2.

We define a generalized infinitely repeated game where the players do not only choose a strategy to

be played for each round (depending on the history of play), but also decide whether the respective

strategies are implemented in short–term contracts or in one–sided long–term contracts. If the latter

option is chosen, the players can deviate from that strategy only to more competitive ones in future

rounds (note that deviations to strategies that the opponent perceives as being more competitive are

supposed to be beneficial to one’s contract partners, as customers or employees).

When we can assume that the equilibrium paths would be “simple” in all subgames (as explained

next), we can show that these generalized repeated games are payoff–equivalent to IMGs (in equilib-

rium). The equilibrium path of a given subgame is understood to be “simple” when there is a unique

stage game strategy profile that is played infinitely often along this path (this profile would be called

“focus point” of the respective subgame). Note that we do not require any kind of consistency of the

focus points over different subgames. Mainly, we want (and need) to rule out circular equilibrium

paths, as those can not be replicated in IMGs; but when they are ruled out, all IMG equilibria have

corresponding equilibria in generalized repeated games, and vice versa.

2

In some circumstances (as infinitely repeated Prisoner’s Dilemmas), the assumption of simple

paths appears rather demanding, since the set of payoff profiles that can be sustained in equilibrium

shrinks dramatically. More generally, though, the stage games would have arbitrarily large strategy

sets (e.g. in Bertrand/Cournot competition, contests, public–good contributions), and then, all payoff

profiles in non–simple paths can be approximated in simple paths. If, furthermore, short–term adap-

tations of the strategies would be associated with costs, then all non–simple paths would be Pareto

dominated by nearby simple paths. In these circumstances, the equilibria in simple paths are even

unique.

We interpret opportunities to sign long–term contract as means to(re)negotiatethe payoff allo-

cation to be sustained along the equilibrium path. Short–term moves, to contrary, allow to defend

long–term allocations against short–term deviations (through temporary retaliations of such devia-

tions). This distinction suggests thatrenegotiation proofnesswould concern only the Pareto effi-

ciency of the long–term commitments in this context. Consequently, short–term retaliations need

not be Pareto efficient. We model that through applying the limit–of–means criterion in the case of

generalized repeated games (i.e. through assuming that the players do not discount future payoffs),

thanks to which temporary retaliations are payoff irrelevant in the long term (but note that even under

the discounting criterion, temporary retaliations are only marginally inefficient).

Given that we apply the limit–of–means criterion, it is even more significant that (for a wide

range of interactions) the payoff–uniqueness of the IMG equilibria continues to hold in general-

ized repeated games (e.g. in all Bertrand competitions, public–good contributions, and sufficiently

symmetric Cournot competitions and contests). For, the combined assumptions of simple paths and

renegotiation proofness merely discretize the sets of payoff profiles that can be sustained in equilib-

rium, but these sets can still be arbitrarily large (they are bounded only by the number of individually

rational and Pareto efficient stage game strategy profiles, which itself is unbounded). Thus, the reduc-

tion of the set of equilibria observed in our model stems from allowing the players to make long–term

commitments, which appears straightforward in a number of circumstances.

Finally, we define a model of repeated games where (for each round) an opportunity to sign a

long–term commitment arises only with some probabilityπ > 0. Here,π is allowed to be arbitrarily

small, and the resulting model is calledperturbed infinitely repeated games. We find that, regardless

of how smallπ is, the allocations that can be sustained in equilibrium are equivalent to those of gener-

alized repeated games, and hence to those of irreversible move games. Thus, marginal perturbations

of infinitely repeated games suffice to obtain the uniqueness of the outcomes that we find for a wide

range of IMGs. This appears interesting, as most other concepts struggle to refine the equilibria of

repeated games.

In Section 2, we shall provide a number of introductory examples (one for each of the most

prominent competitive games) that illustrate our model, our assumptions, and the main aspects of the

equilibrium induction. In this context, we also discuss the various links to previous studies of dy-

3

namic competition and irreversible moves. The two main strings of studies on irreversibilities shall

be introduced already here, however. As mentioned, one of these strings includes studies of finite

irreversible move games with exogenous time horizon Romano and Yildirim (2004) and Yildirim

(2004). We complement these studies by providing the results for several kinds of infinite interac-

tions. Moreover, Feuerstein and Gersbach (2004) study irreversible investment in infinite interactions

and describe equilibria in grim strategies. They show that for discount factorsδ near 1, collusive al-

locations can be sustained in equilibrium. As a complement to that, we showwhichequilibria result

(under renegotiation proofness) in a wider class of games, and how this can be generalized to games

with voluntary irreversibilities. Note also that Feuerstein and Gersbach (2004) survey a number of

additional empirical and theoretical studies on irreversibilities in industrial interactions.

On the other hand, there are studies of irreversible move games (e.g. Lockwood and Thomas,

2002) where precisely the opposite direction of irreversibilities is assumed. Thus, alternative de-

signs of public–good contribution mechanisms are analyzed. The applications of these models (de–

escalation mechanisms) differ from those of our model (strategies to prevent escalation). However,

the equilibrium outcomes in our case generally Pareto dominate theirs, which suggests that it is easier

to prevent escalation than to de–escalate conflicts. This is discussed in detail below (Section 2).

In Section 3, the competitive games and based on them irreversible move games are defined, and

their equilibrium paths are analyzed. In Section 4, we show that irreversible move games are (payoff–

) equivalent to generalized repeated games and perturbed repeated games. Section 5 concludes, and

some of the proofs are relegated to the appendix.

2 Some Introductory Examples

In this section, we give a number of examples that introduce our approach to irreversibilities and

infinitely repeated games intuitively, and that allow us to differentiate our model from previously

published ones. For most of this section, we focus on irreversible move games (IMGs), whereas

the relation of IMGs to generalized or perturbed repeated games is described below. To follow the

arguments in this section, the definitions of “competitiveness” and IMGs as they were provided above

(in the introduction) are sufficiently precise, but note that formal definitions are provided in Section

3.

Further terms that we use (and that are defined formally in Section 3) arestage strategyand

endpoint. We formalize the definition of IMGs by defining a tree of stages, and each stage is defined

through the history of previous strategy adaptations. In each stage the players move simultaneously,

and the sets ofstage strategiesare the strategies of the constituent game that can be played in a

given stage. The path of play is the sequence of stage strategies that is implied under a given IMG

strategy profile, and itsendpointis the profile of stage strategies that is played finally. In Section

4

4, we introducefocus pointsas the corresponding profiles in repeated games: the focus point is the

(assumingly unique) strategy profile that is played infinitely often along the equilibrium path of a

given subgame. Under the limit–of–means criterion, the focus points are uniquely payoff–relevant

in given subgames of repeated games. Finally, let us clarify that when we refer to renegotiation

proofness, we understand it in the sense of Pareto perfectness (Bernheim et al., 1987): the strategy

profiles are to induce Pareto efficient Nash equilibria in all subgames.

2.1 Repeated Prisoner’s Dilemma

In Prisoner’s Dilemmas (PDs) there are only two choices, cooperation (c) and defection (d), and we

defined to be more competitive thanc. In the following, the payoff relations are understood in one

of the usual ways, e.g. 3,2,1,0. In IMGs, the unique Pareto perfect equilibrium implies(c,c). For,

any deviation from(c,c) would result in the endpoint(d,d), and thus, a coordinated deviation from

(c,c) is Pareto dominated, and unilateral deviations imply losses to the deviating players.

In an infinitely repeated game with arbitrarily small time preferences (or under the limit–of–

means criterion), the Folk theorem of Fudenberg and Maskin (1986) applies. Thus, for each of the

payoff allocations indicated in Figure 1 there is a subgame–perfect equilibrium (SPE) where the

respective allocation describes the mean payoffs along the equilibrium path (notably, in this figure

we allow for circular paths). There are four distinguished points in Figure 1.A would result from the

permanent play of(c,d), B from (c,c), C from (d,c), andD from (d,d).

When we restrict the attention to simple paths, only the payoffs associated withB andD can

be sustained in equilibrium. When we assume Pareto perfectness and allow for circular paths (of

PDs), we find that all outcomes on the Pareto frontier can be sustained (Farrell and Maskin, 1989;

technically, these equilibria are shown to be strongly perfect in the sense of Rubinstein, 1980, and

hence they are Pareto perfect). When we assume simple paths and Pareto perfectness, the unique

corresponding outcome implies the permanent play of(c,c).

Thus, it appears that the assumption of simple paths is already sufficient to obtain the equivalence

Figure 1: Possible Equilibrium Payoffs (in Circular Paths) in Infinitely Repeated PDs

- profit 1

6

profit 2

rD

rA

rC

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5

of repeated games and IMGs. This is true in the case of PDs, but it is not generally. When there are

several Pareto efficient and individually rational stage game strategy profiles, then there are also

several renegotiation–proof equilibria in simple paths—but only one them corresponds with an IMG

equilibrium. Here, it is relevant whether the players are provided with opportunities to restrict their

strategy sets (however unlikely these opportunities are). In these circumstances, the perturbation of

repeated games provides highly significant refinements upon renegotiation proofness (and even upon

renegotiation proofness in combination with simple paths).

Secondly, we see that the assumption of simple paths is necessary. For, players who can restrict

themselves in repeated games would not restrict themselves to playd in the above (circular–path)

equilibria. Such a restriction would imply the permanent play of(d,d), which is neither individually

rational nor Pareto efficient in relation to the paths in non–simple equilibria. Generally, the assump-

tion of simple paths is not as restrictive as in PDs, however, as most payoff allocations of non–simple

paths can be approximated in simple paths.

In this context, let us relate our model of IMGs to that of Romano and Yildirim (2004). They

define an IMG with an exogenously fixed and finite number of stages (T), whereas in our case, the

number of stages played is endogenous (and indefinite). Thus, there is a last period in their model, and

in that last period, the players would play a strategy profile that is a Nash equilibrium within the sets of

remaining strategies (this feature allows the play to be induced backwardly). For instance, the unique

outcome in a Prisoner’s Dilemma would be(d,d), regardless of what the number of rounds is. In our

model, to the contrary, there is no last stage where(d,d) must necessarily be played, which allows the

players to collude—leading to the unique outcome of(c,c) in PDs. This intuition generalizes to all

competitive games: in finite IMGs, the equilibria imply either the Nash outcome of the stage game, a

Stackelberg outcome, or an intermediate outcome (partially Nash, partially Stackelberg), whereas in

typical applications of infinite IMGs the unique equilibrium implies collusion at (or, near) the cartel

strategies.

2.2 Public Good Games

Next, let us consider irreversibilities in infinitely repeated public–good games. In a public–good

game, the players can contribute amounts as they are described in their strategy setsSb = {0,1, . . . ,n},and the payoff ofb is Ub(s1,s2) = δ∗ (s1 +s2)−sb. In this context, a strategysA

b is understood to be

more competitive thansBb if it implies a smaller contribution. In turn, this implies for IMGs that the

players can only deviate from their last stage contribution by decreasing it. For instance, a fisher can

enlarge his fleet, but he can not reduce it.

Apparently, under renegotiation proofness, contributions of(n,n,) must result along the equilib-

rium path. Precisely, when the players have played(1,1) in the previous stage, the situation is equiv-

alent to a PD, and further restrictions are Pareto dominated (and even weakly dominated). Hence,

6

(1,1) is an endpoint, and there is generally a player who would deviate from points as(1, i) with

i > 1. As a result, no player would deviate from restrictions to(2,2), since deviations would trigger

the endpoints(0,0) or (1,1). In this way, the backward induction can be completed to show that

(n,n) is the unique endpoint of equilibrium paths. Thus, Pareto efficiency is generally and uniquely

secured.

To the contrary, in plain repeated games, the set of contributions sustained in renegotiation–proof

equilibria includes all profiles(i,n) with i ≤ n that are individually rational. Thus, the outcomes of

repeated games differ significantly from those of IMGs (under renegotiation proofness) even if we

restrict the attention to simple paths. Since the equilibria of IMGs are payoff–equivalent to the equi-

libria of perturbed repeated games, this shows that introducing possibilities to restrict one’s options in

repeated games can have significant strategical implications. Such refinement effects can be observed

whenever there are several pure strategy profiles in the constituent game that imply Pareto efficient

payoff allocations.

Our model of public good games complements previous studies of irreversibilities in public

good games, e.g. Admati and Perry (1991), Marx and Matthews (2000), Gale (2001), and in the

most general way in Lockwood and Thomas (2002). In these studies, precisely the opposite direc-

tion of irreversibilities had been assumed, i.e. the players had been allowed only to increase their

contributions in the course of the game. The applications of these models include (Lockwood and

Thomas, 2002) donations to funds, pollution reductions, capacity reductions in a declining industry,

disarmament, and tariff reductions. Apart from fund raising, the studied models are understood as

mechanisms to de–escalate conflicts. To the contrary, we study circumstances where the players are

best off preventing the conflict from escalating in the first place.

In relation to the outcomes that result in our models, their outcomes are generally not efficient

(neither Pareto efficient nor socially efficient). Apparently, when players can only reduce their con-

tributions (as in our model), the players are generally best off imitating their opponent’s behavior

(i.e. to reciprocate negatively). However, when the contributions are only to be increased, imitating

the opponents is not generally equilibria. In fact, where one would contribute is largely independent

of how much the opponent has contributed before; mainly, it depends on whether one’s contribution

would trigger further contributions of the opponent. Generally, the latter is not satisfied at the ends

of the game tree, i.e. in subgames with previous contributions of(n−1,n−1), and iteratively, this

can prevent contributions to be made altogether.

2.3 Symmetric Competition

The next examples concern irreversibilities in models of Bertrand and Cournot competition. First,

we shall examine a model of competition where the consumers react symmetrically to decisions of

7

the duopolists. That is, the payoff functions are assumed to satisfy for all playersb

Ub(sb,s−b) = sb∗ (1−sb +αs−b) with |α| ≤ 1. (1)

Notably, if we understand thesb to represent prices, a Bertrand model results, and if they represent

quantities, a Cournot model results. Generally, the strategical issues that arise are independent of the

paradigm that we assume: n a model withα > 0, the strategies are complementary (in the sense of

Bulow et al., 1985), and in a model withα < 0, the strategies are substitutionary. However, ifα > 0

and we understand thesb as prices, the underlying goods are substitutes of the consumers’ eyes, but

if α > 0 and we understand thesb as quantities, the underlying goods are complements. In the case

of quantity competition,α = 1 implies (as a limiting case) the Cournot model of homogenous goods.

The best–reply functions, the Nash strategiessNb ≡ s∗b, and the cartel strategiessM

b are

Rb(s−b) =1+αs−b

2, sN

b =1

2−α, sM

b =1

2−2α. (2)

We see thatsNb < sM

b ⇔ α > 0. Thus, in the case of Bertrand competition andα < 0, the customers

would benefit from tacit collusion (towards the cartel prices), as it implies decreasing prices. Like-

wise, the customers benefit under Cournot competition andα > 0 (as the quantities increase), and

generally, they benefit in cases of complementary goods. Moreover, forα → 1, the cartel strategies

converge to infinity, i.e. the players would collude at nearly infinite prices or quantities.

Again, we define IMGs as games where adaptations of the stage game strategies are possible only

towards more competitive strategies. In this case, the relation “more competitive” is defined through

the relation of Nash and cartel strategies, with the Nash strategies being competitive and the cartel

strategies being collusive. Thus, ifsNb < sM

b , then a strategysAb is more competitive thansB

b iff sAb < sB

b.

As a result, the strategies may only be decremented in the course of the interaction. Alternatively, if

sNb > sM

b , then the strategies may only be incremented. Note that in the case of complementary goods,

these directions are somewhat unintuitive (increasing prices, decreasing quantities), but generally, we

would assume that competitors produce rather substitutionary than complementary goods.

Since our strategy sets are assumed to be discrete, all strategies can be represented asskb = sN

b +kεfor some sufficiently smallε > 0. The cartel prices correspond withK = ε−1∗α

2(1−α)(2−α) and k = 0

represents the competitive (Nash) strategies. Apparently, ifα > 0 thenK > 0 and if α < 0 then

K < 0. To simplify the notation, let us define the range(0, . . . ,K) to be understood as(K, . . . ,0) in

caseK < 0.

Our backward induction of the play in the irreversible move game starts ink = 0 and ends

in k = K (which is the opposite direction of the actual irreversibilities). To simplify the backward

induction, let us assume in this section that all strategiesskb that can be defined through ak∈ (0, . . . ,K)

are indeed in playerb’s strategy set. Thus, the following identity holds for both playersb∈ B.{sb ∈ Sb : sN

b ≤ sb ≤ sMb

}={

s∈ IR : ∃k∈ (0, . . . ,K) , s= sNb +kε

}. (3)

8

Figure 2:Strategic complements(i.e. α > 0, left) andStrategic substitutes(α < 0, right) End-

points in quasi continuous strategy spaces. In the case of strategic substitutes, it is supposed that (17)

applies.

-

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As a result of the second assumption, the above game qualifies assimply structured competitive(for

a general definition see 3.16), and otherwise it would qualify ascompetitive(in the sense of 3.1).

In simply structured games, the path(skb,s

k−b) for k = 0, . . . ,K is between the best reply curves (14),

along it both players’ payoffs are increasing monotonically (15), and the payoffs associated withK

are Pareto efficient with respect to all possible payoff profiles (16).

In the unique renegotiation–proof SPE of the respective IMG,(sKb ,sK

−b) results (see also Figure

2), and all points with(skb,s

k−b) for k= 0, . . . ,K are (potential) endpoints. Since traversing along these

points fork = 0, . . . ,K implies Pareto improvements, the players are generally worse off initiating

deviations from any of these points towards a point nearerk = 0. The respective backward induction

starts ink= 1, where deviations tok= 0 (or arbitrary more competitive points) are Pareto dominated.

As a result,(skb,s

k−b) for k = 1 is an endpoint, deviations fromk = 2 would be Pareto dominated, and

so on.

Note that ifα < 0, the competitiveness of the best replies is decreasing in the competitiveness of

the opponent’s strategy. Thus, the best response to some opponent’s strategy that is less competitive

than his Nash strategy (e.g. to somek < 0) is a strategy that is more competitive than one’s Nash

strategy (i.e. with somek > 0). Generally, points(sA,sB) such thatsA ≥ RA(sB) andsB ≥ RB(sA) are

endpoints, i.e. when they are reached in IMGs, the players would not further adapt their strategies.

If α < 0, such points can be reached through unilateral deviations from anysk. In Lemma A.1 we

see that no player is better off deviating unilateral from somek ∈ (K, . . . ,0) to such an endpoint in

the above model of competition. In simply structured games in general, we assume that no player

would be better off deviating in this way (17). In our most general framework, competitive games,

we analyze the play without any such assumption.

9

In the limiting case of homogenous Cournot competition (α =−1), the players are merely indif-

ferent with respect to deviations from thesk to such extremely competitive endpoints (i.e. to endpoints

where his strategies are more competitive than his Nash strategy). Thus, the collusive endpointssk

are merely weak, but they are uniquely renegotiation proof. In equilibrium, the cartel pricessK re-

sult uniquely and the cartel surpluses are shared equally amongst the players. Let us note again

that this uniqueness provides significant refinements upon renegotiation proofness in plain repeated

games. For, even under strong perfectness and the discounting criterion, asymmetric allocations are

not ruled out in equilibrium (precisely, the players are secured only to get a quarter of the monopoly

payoff, Farrell and Maskin, 1989).

2.4 Asymmetric Competition

Briefly, let us generalize the previous example to an asymmetric family of models, such that for all

playersb

Ub = sb∗ (1−sb +αbs−b), with |αb| ≤ 1, αb∗α−b ∈ (0,1]. (4)

Basically, this analysis follows the guidelines of the previous one. Thanks to the increased generality,

however, we can now point out the potential obstacles to tacit collusion in IMGs more precisely.

Playerb’s best reply function, his opponent’s inverted best reply, his Nash strategysNb , and his cartel

strategysMb , which happens to equalsM

−b, are

Best replies: Rb(s−b) =12

(1+αbs−b) R−1−b(s−b) =

1α−b

(2s−b−1) , (5)

Equilibria: sNb =

2+αb

4−αbα−bsMb =

12−αb−α−b

. (6)

Note that

∀b pNb =

2+αb

4−αbα−b<

2+αb

4−αbα−b−2α−b−α2b

= pMb

⇔ α−b >−α2

b

2and αb >−

α2−b

2⇔

αb∗α−b>0αb,α−b > 0 (7)

As before, we assume that both Nash and cartel strategies are inb’s strategy set and that his strategy

set is discrete. Thus,b’s collusive strategiesskb ∈[sNb ,sM

b

]correspond with somek∈ (0, . . . ,K) using

K = (sMb − sN

b ) ∗ ε−1, such thatskb = sN

b + kε. Moreover, we assume again that for allk ∈ (0, . . . ,K)

the corresponding strategyskb is indeed inb’s strategy set (thus, the game might qualify as simply

structured).

For αb,α−b > 0 (strategic complements), the case is essentially equivalent to the previous one.

We can backward induce the play, and the players collude at(pK

b , pK−b

)≡(pM

b , pM−b

).

Obstacles to tacit collusion can arise ifαb,α−b < 0. Playerb is better off deviating unilaterally

from (pkb, pk

−b) to a strategy beyond his Nash prices, i.e. toR−1−b(pk

−b), if and only if αb < 0 and

10

Figure 3:Asymmetric competition The set of parameter profiles compatible with tacit collusion

- α1

6

α2

−1

−1

�

����

�������

��������

��������

��������

�����

��

����

�����

����

����

αb +α2−b > 0 (Lemma A.2). Likewise, his opponent is better off iffα−b +α2

b > 0. In the symmetric

caseαb = α−b (see above), none of these inequalities had been satisfied, but here, they may be

satisfied, and as a result the set of collusive endpointssk < sN may be empty. The region in the

parameter space where collusive outcomes are supported is illustrated in Figure 3, and in all of these

cases, the cartel prices result uniquely.

Besides our relation to the study of Farrell and Maskin (1989), let us relate our outcomes also to

the study of Maskin and Tirole (1987, 1988). They analyzed the implications of asynchronous moves

in combination with Markov strategies in some repeated oligopoly games (the respective equilibria

are called Markov perfect). In general, the assumption of asynchronous moves helps the players to

coordinate towards Pareto efficient equilibria (Lagunoff and Matsui, 1997), i.e. it implies renegoti-

ation proofness. In combination with Markov strategies (i.e. with strategies that are functions only

of the previously played strategy of the opponent), however, this effect is sacrificed to the favor of

strategical simplicity. Interestingly, this simplicity yields unique equilibria in a number of circum-

stances, but it rules out temporary retaliations of aggressive strategies. The latter complicates the

sustainment of collusive outcomes in equilibrium; in repeated PDs, for instance, the unique equilib-

rium implies permanent defections. Besides this simplification, the assumption of Markov strategies

offers an interpretation that we shall present below.

Similar to the case of PDs, the Markov perfect equilibria of repeated Cournot duopolies are not

collusive; but here, they are even more competitive than (and Pareto dominated by) the constituent

game’s Nash equilibrium. As described above, in our model the cartel strategies result. Maskin and

Tirole have not analyzed Bertrand competition with heterogenous goods explicitly, but it appears

that the Cournot results extend equivalently to the Bertrand case, and thus, the outcomes are never

collusive. In relation to our results, these equilibria appear somewhat unintuitive.

11

Figure 4: The situation in a rent–seeking contest

-

6

xA

xB

rr��

��

��

��

��

��

��

��

��

��

�

��

���

��

��

��

��

��

��

��

��

��

��

�

��

���

��

��

��

�

���

��

xmin

xmin14

14

1

1

RB(xA) =√

xA−xA

RA(xB) =√

xB−xB

QQQk

�������

endpoints

2.5 Contests

Two competitorsA andB can choose expendituressb,s−b ∈ {smin,smin + ε,smin + 2ε, . . . ,smax} in

order to win a price which is valued 1 by both players (a case of a contest with asymmetric prices is

analyzed below). The probability thatb wins the price is Probb = sb(sb + s−b)−1 andb’s expected

payoff is

Ub =sb

sb +s−b−sb. (8)

The best reply functions are (in a quasi–continuous case)Rb(s−b) = s0.5−b− s−b ∀b, and the unique

Nash equilibrium issNA = sN

B = 14.

Strategically, contests are neither purely complementary nor purely substitutionary (see also

Figure 4). However, it can be shown rather easily that contests (as described above) qualify as simply

structured competitive for smallε. This implies that no player is better off deviating unilaterally to a

strategy beyond his Nash strategy in the sense of Equation (17), which is shown in Lemma A.3. Thus,

we can backward induce (as above) that the set of collusive endpoints is{(sA,sB) : sA = sB ≤ 1

4

}, and

that the smallest possible contributions(sminA ,smin

B

)result uniquely along the equilibrium path.

Now, let us generalize this to a contest with asymmetric valuations. The price of the contest

is worth PA for A and PB for B. To circumvent some cases that need technically more involved

treatments, let us assume thatP2b +1≥ P−b ∀b. For instance, this is satisfied in all cases with1

2P−b≤PB ≤ 2P−b ∀b, i.e. if the players do not value the price too differently. We shall describe a simply

structure instance of such a contest (generalizations can be treated similarly). The expected profits

and best reply functions are

Ub(sb,s−b) = Pb∗sb

sb +s−b−sb and Rb(s−b) =

√Pbs−b−s−b. (9)

12

Thus, the one–round Nash equilibrium strategies aresNb = cPb with c = PAPB ∗ (PA + PB)−2. The

dynamic contest is simply structured, if (for instance) the strategies can be described asin ∗sN

b , i.e. if

the strategy sets satisfy

Sb ={

r ∈ (0,Pb) : ∃i ∈ IN, r =in∗cPb

}.

We refer to the strategies of playerb assib = i

ncPb. Moreover, Equation (17) is generally satisfied for

i ≤ n andP2b +1≤ P−b (Lemma A.4). The set of potential endpoints implying tacit collusions≤ sN

is equivalent to{(sA,sB) ∈ (0,Pb)2 : ∃i ≤ n,∀b, sb =

in

cPb

}, (10)

and thus(1

ncPA, 1ncPB

)is the unique outcome of renegotiation–proof SPEs in IMGs.

There are several previous studies of dynamic contests. Most of these studies deal with two–

stage contests (Dixit, 1987; Baik and Shogren, 1992), and as a result, the induced dynamics are rather

different from those in our model. An exception is the study of Leininger and Yang (1994), who

analyze infinite–round contests and describe an equilibrium in Markov strategies and asynchronous

moves. According to this equilibrium, the firstly moving player invests half of the eventual price,

and the secondly moving one invests nothing. The dynamics induced through the Markov strategies

imply monotonically increasing expenditures off the equilibrium path, which therefore resembles our

model. Apparently, the main difference from our model (yielding the asymmetric outcome) concerns

the asynchronous bid structure. The described equilibrium is socially inefficient and, more strikingly,

it is not Pareto–superior to the Nash equilibrium of the respective one–round contest. However,

it is not Pareto–dominated by our equilibrium (contrary to Markov perfect equilibria in the above

cases of Bertrand/Cournot competition). Finally, let us refer to Yang (1993), who finds in different

circumstances that equilibria implying minimal expenditures can occur (but not uniquely so, contrary

to our results).

3 The Analysis of Irreversible Move Games

In this section, we define competitive games as two–players games with simultaneous moves, a dy-

namic game based on this class of games, and determine its equilibria. The dynamic game that we

analyze is called irreversible move game (IMG), and it serves as an abstract representation of the

strategical constellation in a number of alternative dynamic games that are introduced in the next

section.

13

3.1 Competitive Games

There are two players,A andB, and they play a simultaneous–move game with finite strategy spaces

SA andSB. Their payoff functions areUA,UB : SA×SB → IR. Let us defineS= SA×SB and the

payoff profile isU(s) =(UA(s),UB(s)

). For each player’s set of strategies we assume an ordering>

(which is transitive and complete), and this ordering is referred to asmore competitive. Based on this

ordering, we define several restrictions of the valuation functions, and the game is calledcompetitive

if all of them are fulfilled. In the following, a representative player is calledb, his strategies are

sb ∈ Sb, and his opponent’s strategies ares−b ∈ S−b.

Definition 3.1 Competitive Gamessatisfy the following characteristics (i), (ii), (iii).

(i) CompetitivenessThe more competitive the strategy, the lower the opponent’s payoff, i.e. for

all b, sb, s−b, s′−b : s−b > s′−b ⇒ Ub(sb,s−b) < Ub(sb,s′−b).

(ii) Unimodality of Partial Payoff Functions The utility functions are strictly unimodal in the

competitiveness of the own strategies and for each opponent’s strategy there is a unique best

reply. The best reply function isRb(s−b) ∈ argmaxsb Ub(sb,s−b), and the joint best–reply

function is calledR(s) := (RA(sB),RB(sA)). Formally, the payoff’s unimodality implies for all

s−b ∈ S−b and allsb 6= s′b ∈ Sb : sb≶ s′b≶ Rb(s−b) ⇒ Ub(sb,s−b)≶Ub(s′b,s−b).

(iii) Intersections of Best RepliesThe continuous functions resulting from linearly intrapolating

the (discrete) best reply functions have a unique intersection, which is in the strategy profile

s∗ ∈ S. Necessarily,s∗ is the unique Nash equilibrium in pure strategies, and for allsbR s∗b we

havesbR Rb(R−b(Sb)

).

Note that we did not assume the best reply functions to be monotonic in any sense. However, in the

literature (Bulow et al., 1985), games with best reply functions that are increasingly competitive in the

opponent’s competitiveness are said to be strategically complementary, and games with decreasing

best reply functions are said to exhibit strategical substitutes. Next let us introduce some notation

which simplifies the following analysis.

Definition 3.2 Vector relations In the following,s≤ s′ shall be understood as being equivalent to

sA ≤ s′A andsB ≤ s′B. Similarly, The strict relations< s′ is equivalent tos≤ s′,s 6= s′. The relations

�,� refer to the logical negations of the above statements. Finally, defines� s′ as being equivalent

to sA < s′−A andsB < s′B.

In this sense, we can define the maximum of a set of strategy profilesS′ ⊆ S as the set of profiles

maxS′ such thats∈maxS′ is equivalent tos∈ S′ and there exists nos′ ∈ S′ that satisfiess′ > s,

maxS′ ={

s′ ∈ S′ : ∀s′′ ∈ S′, s′ ≮ s′′}

. (11)

14

The minimum ofS′ is defined correspondingly. Finally, let us define theinterior of the strategy set

as the set of strategy profiles that are less competitive than the joint best replies.

Definition 3.3 The interior SI ⊆ Sof the strategy space isSI = {s∈ S: ∀b,sb ≤ Rb(s−b)}.

3.2 Irreversible Move Games

IMGs are multi–stage games where adaptations of the stage game strategies are restricted to be

weakly increasingly competitive. Moreover, IMGs end when a stage game strategy profile is played

repeatedly and only these finally played strategies are payoff–relevant. Each stage is described

through a history of play, and the set of all stages is

K ={(s0,s1, . . . ,st) : t ≥ 0, s0 ∈minS, ∀t ′ ≤ t : st ′ ∈ Sandst ′ > st ′−1, st ≥ st−1

}. (12)

A stage isterminal if st = st−1, and it is adecisionstage otherwise. The payoff profile induced in

terminal stages isU(st), and the set of terminal stages is calledKT . The set of strategies in decision

stages is the set of strategies that lead to further stages, i.e.{s : s≥ st}, and the set of decision stages

is calledKD.

An IMG strategy is a function describing a feasible action for each decision stage ˆsb : KD → S,

ands= (sb) denotes an IMG strategy profile. Since we restrict our attention to pure strategies, the

terminal stage that follows a given decision stage under some ˆs is unique. The last component of the

history in a terminal stage is calledendpoint, and the endpoint functiones : KD → S describes for

each decision node the endpoint that is induced under a given profile ˆs. The set ofpotentialendpoints

Es induced in a strategy profile ˆs is the set of stage game strategy profiless for which there exists a

decision stagek∈ KD such thats= es(k) is the resulting endpoint,

Es= es(KD) ={

s∈ S: ∃k∈ KD, s= es(k)}

. (13)

Next, let us describe the solution concept that we apply. We assume that the IMG strategy pro-

files under consideration are renegotiation–proof subgame–perfect equilibria in the following sense

(Pareto perfectness). Thus, we rule out threats of inefficient strategies off the equilibrium path. First,

we define the Pareto relations.

Definition 3.4 Pareto RelationsA strategy profiles′ ∈ SPareto dominatess′′ ∈ S iff U(s′) > U(s′′).

A strategy profiles is Pareto efficient with respect to a setS′ ⊆ S iff for all s′ ∈ S′ : U(s′) ≯ U(s).

Definition 3.5 Pareto Perfect Equilibrium (PPE) The strategy profile induces Pareto–efficient

Nash equilibria in all subgames (Bernheim et al., 1987).

Additionally, let us impose an assumption that simplifies some of the notation but is irrelevant oth-

erwise. In particular, it does not affect the equilibrium payoffs, i.e. we could assume precisely the

opposite direction without affecting the equilibrium payoffs.

15

Assumption 3.6 If two endpoints induce equivalent payoffs and if one of them is less competitive

than the other one, the equilibrium path implies the more competitive one.

3.3 Characteristics of Pareto–Perfect Equilibria (PPEs) in IMGs

In order to get a basis for backward inductions in IMGs, we shall first determine the endpoints that

are not below the Nash equilibriums∗. These endpoints are described in the following Lemma (its

proof is relegated to the appendix).

Lemma 3.7 Consider an arbitrary PPE ˆs of an IMG. A points≮ s∗ is an endpoints∈ E induced in

s if and only if s≥ R(s). The endpoints∗ Pareto dominates all endpointss′ ≥ s∗.

Thanks to Lemma 3.7, we know all endpointse that satisfye� s∗. In the following, we shall

backward induce the remaining endpoints.

Proposition 3.8 In any PPE of an IMG, the induced set of potential endpointsE satisfies for alls∈S

thats∈ E if and only if s Pareto dominates all points inEs = {e∈ E : e> s}. Thus,E is unique.

PROOF. The “if” part of the proposition follows directly from the characteristics of Pareto–perfect

equilibria. To prove the “only if” part, we shall contradict the opposite hypothesis. Assume there is a

PPE such that there are endpointss∈E that do not Pareto dominate all points inEs = {e∈E : e> s}.Let Econ denote the set of alls∈ E that contradict our claim, and consider anysc ∈ maxEcon. As a

result, the above proposition must be satisfied for alls′ > sc. Since it is not satisfied forsc, however,

there is somee∈ E : e> sc that is not Pareto dominated bysc, and hence, there is a playerb such that

Ub(e) > Ub(sc).

Whenb deviates unilaterally toeb, one of the Pareto efficient endpoints inE′ = {e′ ∈ E : e′ ≥(eb,sc

−b)} would result. Generally, playerb is better off in(eb,sc−b) than ine, assc

−b < e−b. All of the

Pareto efficient points ine′ 6= e∈ E′ satisfy (by assumption) thate′ ≯ e, and thuse′−b < e−b. Thus,

in any continuation equilibrium following a stage with(eb,sc−b), b’s opponent plays a stage strategy

s−b ≤ e−b. Hence, no such continuation equilibrium can imply a payoff allocation whereb is worse

off than ine, asb could deviate unilaterally to a strategy where he does not move beyondeb. As a

result,b can secure the payoff resulting ine, and ansc must not be an endpoint. QED

Corollary 3.9 There is a unique set of endpointsE that is induced in all PPEs of a given IMG.

The uniqueness of the set of endpoints does not generally imply that the equilibrium paths in all

subgames are unique. Consider a stagek ∈ KD with st = s, define the set of endpoints that can still

be reached asEs{e∈ E : e≥ s} . Under renegotiation proofness, the continuation equilibrium must

lead to one of the Pareto efficient points inEs. Thanks to the construction ofE (and thusEs), this

implies that the subgame’s equilibrium path ends in one of the points in minEs.

16

Lemma 3.10 Consider a decision stagek ∈ KD and denote the set of potential endpoints in the

resulting subgame asEst = {e∈ E : e≥ st}. Any PPE path in that subgame must imply an endpoint

in minEst .

Thus, a subgame’s equilibrium path is unique (apart from payoff–equivalence and interchangeability)

iff minimum of Est is unique. The uniqueness is examined in the following. First, consider the

following lemma.

Lemma 3.11 For all pairs of endpointse1 6= e2 ∈E : e1,e2 < s∗, eithere1 > e2 or e2 > e1 is satisfied.

Moreover, all endpointse< s∗ are interior, i.e. they satisfye≤ R(e). (The proofs are given in the

appendix.)

As a result, the set of endpoints that are less competitive thans∗, E′ := E∩{s∈ S : s≤ s∗}, can

be completely ordered by “�”. That is, we can connect all of these endpoints through a (unique)

monotonically increasing line (see the examples in Section 2). In some sense, this line characterizes

the strategical aspects of IMGs. This leads straightforwardly to the following.

Lemma 3.12 There is generally a unique endpoint that can result along the path of a PPE where

no player is worse off than in the constituent game’s Nash equilibriums∗ (the proof is given in the

appendix).

As a result, if for all endpointse∈E : e≮ s∗ we havee≥ s∗ (as in the case of strategic complements),

the endpoint that results along the equilibrium path is unique, and no player is worse off than ins∗.

Next, let us illustrate the relative equilibrium payoffs if the Pareto efficient endpoint is not unique.

Lemma 3.13 Let EPs denote the set of potential Pareto–efficient endpoints in a stagek ∈ KD with

st = s. For alle′ 6= e′′ ∈ EPs , we have for allb thate′b > e′′b ⇔ Ub(e′) > Ub(e′′) (the proof is given in

the appendix).

This observation allows us to establish the payoff–equivalence of IMGs and generalized/perturbed

repeated games (see below). In the following, we shall use it to illustrate the assumption of alternating

moves in IMGs, instead of Pareto perfectness. Here, we shall assume that for alls′ 6= s′′ ∈ S in the

constituent game and for all playersb, the induced payoffs are differentUb(s′) 6=Ub(s′′). Generically,

this is satisfied. Thus, it is straightforward to induce (backwardly) that the sets of potential endpoint

E is equivalent to that under Pareto perfectness and that Pareto–efficient paths result in all subgames.

More significantly, as a consequence of Lemma 3.13 we have for all playersb, all s∈ S, and

all stagesk∈ KD : st = s, that if e∈ Es denotes the (necessarily unique) endpoint that maximizesb’s

payoff, thene is also the unique Pareto efficient endpoint in a subgame following the play(eb,s−b).

As a result, the first moving player in a alternating–move IMG can secure the endpoint that maximizes

his payoff, and thus, the equilibrium outcome is unique.

17

Proposition 3.14 Generically, in IMGs with alternating moves, the subgame–perfect equilibrium

outcome (payoff allocation) is unique. (Note that we need not assume Pareto perfectness.)

3.4 Simply Structured Competitive Games

If the stage game is (beyond its competitiveness) simply structured in the sense defined below, the

endpoints are particularly easy to induce. All of the examples that we gave initially are simply

structured in this sense. The simplified way to induced the endpoints is defined first.

Definition 3.15 Simplified Induction Start withs0 = s∗ and construct the pointssk+1 = max{s∈S : s� sk} until the newly constructed point fails to yield Pareto improvements. The last point

implying Pareto improvements is calledsK.

Thus, we construct a sequence of pointssK < · · · < sk < sk−1 < · · · < s1 < s∗ which is connected in

the sense that the deviation from somesk to the next pointsk−1 requires precisely one step per player

(towards a more competitive strategy). This is simplified, we can not go back simply one step per

player in general. Instead, we have to look for the “next” point that Pareto dominates all endpoints

that have already been found.

The set of simply structured competitive games is defined such that all endpointse< s∗ can be

derived in the simplified induction if and only if the IMG is based on a simply structured game. Note

that the remaining endpointse≮ s∗ are described in Lemma 3.7. Thus, when we have shown that a

game is simply structured, all endpoints can be derived easily.

Definition 3.16 Simply structured competitive gameA simply structured competitive game is a

competitive game that satisfies for the(sk)Kk=0 constructed in the stepwise induction

∀k : sk ≤ R(

sk)

(14)

∀k′ < k′′ : U(

sk′)

< U(

sk′′)

(15)

∀b, ∀k, ∀sb > s∗b :(

sb,sk−b

)≥ R

(sb,s

k−b

)⇒ Ub

(sb,s

k−b

)≤Ub

(sk)

. (16)

∀s< sK : Ub(s) ≯ Ub(sK) (17)

∀s : s≥ R(s) ands≯ sK : U(s) < U(sK) (18)

Verbally, in simply structured competitive games, the path fromsK to s∗ lies between the best reply

curves (14), symmetric deviations to more competitive strategy profiles are Pareto dominated (15),

deviations to endpointse≮ s∗ are generally not profitable (16), the players would generally deviate

from s < sK to sK (17), andsK Pareto dominates all potential endpointss≥ R(s) that can not be

reached anymore whensK is reached (18).

18

Proposition 3.17 The “simplified induction” yields the (unique) set of endpoints satisfyinge≤ s∗ if

and only if the stage game is simply structured competitive. Along any equilibrium path,sK results

(the proof is skipped).

4 Dynamic Games that Are Payoff–Equivalent to IMGs

4.1 Infinite Irreversible Move Games

In infinite irreversible move games (IIMGs), stage game strategy adaptations are irreversible as in

IMGs, but the game never stops and all rounds are payoff–relevant. The set of stages in IIMGs is

K ={(s0,s1, . . . ,st) : t ≥ 0, s0 = minS, ∀t ′ ≤ t : st ′ ∈ Sandst ′ ≤ tst ′−1

}. (19)

There are no terminal stages. An IIMG strategy ˆsb maps each stage to a feasible move, i.e. to some

s∈ {s′ ∈ S: s′ ≥ st}. The path of play followingk∈ K according to ˆs= (sb) is (recursively)

p(k) = (pt ′(k))t ′>t using pt ′(k) = s(k, ∪t ′′: t<t ′′<t ′ pt ′′(k)) , (20)

and the endpoint of a subgame is the (unique) stage game strategy profile that is played infinitely

often,

e(k) ∈{

s∈ S: ∃t ′ ∀t ′′ > t ′ s= pt ′′(k)}

. (21)

Based on that, we can define the set of potential endpointsE equivalently to (13). We assume that

the players aggregate the infinite payoff stream according to the limit–of–means criterion (LMC), i.e.

the payoff ofb in the IIMG is

ULMCb = liminf

t→∞

1t ∑

t ′≤t

Ub(pt ′(s0)

). (22)

Thanks to LMC (in combination with Pareto perfectness), the payoff–equivalence of IMGs and

IIMGs is straightforward. That is, only the endpoints are payoff relevant in IIMGs, equivalently

to IMGs, and hence, the backward inductions of the endpoints that would result in given subgames is

equivalent in both games. Likewise, the endpoints that would result in given subgames must be the

Pareto efficient ones in the respective sets of potential endpoints.

Proposition 4.1 The setsE of potential endpoints are equivalent in IMGs and IIMGs, and the end-

points that can result along the equilibrium paths of PPEs are equivalent in all subgames, too.

4.2 Generalized Infinitely Repeated Games

In plain infinitely repeated games (IRGs), the stage game strategies are unrestricted, the game never

ends, and all rounds are payoff relevant. The set of stages isK = {(s0, . . . ,st) : t ≥ 0} and an IRG

strategy ofb is a function ˆsb : K → Sb. The payoff (under LMC) is defined equivalently to (22).

19

We restrict our attention to equilibria insimple paths: for each subgame there is a unique stage

game strategy profile that is played infinitely often. These profiles are calledfocus points, but note

that we do not assume the focus point to be equivalent in all subgames. Using (21) the focus point

along the equilibrium path can be defined asf = e(s0), and likewise, we can define a function that

all maps stagesk∈ K to the resulting focus points. Finally, equivalently to (13) we can define the set

of potential focus points that result according to a strategy profile ˆs in arbitrary subgames.

Apparently, all focus points that induce payoffs which Pareto dominate the minimax payoffs can

be sustained in SPEs. For instance, Rubinstein (1979) and Fudenberg and Maskin (1986) describe

(Folk theorem) equilibria that hold equivalently under the limit–of–means criterion (LMC). More-

over, since temporary retaliations are payoff–irrelevant under LMC, the Folk theorem equilibria are

even Pareto perfect iff all focus points inF are Pareto efficient points with respect toS.

Now we generalize IRGs to generalized infinitely repeated games (GIRGs). Here, the players are

additionally allowed to impose restrictions upon their (own) stage game strategy sets for the stages

following a decision stage. The players can restrict themselves by stating: “In all of the following

rounds, I would play a strategy that is as least as competitive as my current one.” That is, the players

can lock in their current strategies as lower limits for the following stages. When the current stage

game strategy is locked in, a flag is set tof = 1, and otherwise it isf = 0. The set of stages is

K ={(s0, f0,s1, f1, . . . ,st , ft) : t ≥ 0, s0 = minS,

∀t ′ ≤ t : st ′ ∈ S, ft ′ ∈ {0,1}2, st ′ ≥ smin(s0, f0, . . . ,st ′−1, ft ′−1)}, (23)

using

sminb (k) = max{sb ∈ Sb : ∃t ′ ≤ t, st ′,b = sb and ft ′,b = 1}, (24)

as the strategysb ∈ Sb that defines the restriction that playerb has committed to in stagek. Precisely,

he is committed to playsb ≥ sminb (k). The set of stage strategy profiles is the set of profiles that

leads to further stages. A GIRG strategy ˆsb of playerb maps each stage to a feasible stage strategy.

Moreover, for a given GIRG strategy profile ˆs, let us define a path functionp that maps each stage to

the implied path of payoff–relevant moves(st+1,st+2, . . .).

As in IRGs, we restrict our attention to strategy profiles ˆs that induce simple paths. That is, for

all k ∈ K, there is a uniques∈ S that is played infinitely often alongp(k). Again, thiss is called

focus point of the subgame followingk. Let f (k) denote the focus point resulting in the subgame

following k, and equivalently (13), let us defineF as the set of potential focus points. Under LMC,

the payoff in the continuation equilibrium followingk implies the payoff allocationU( f (k)) and the

equilibrium payoff allocation isU( f (s0)).

20

4.3 Payoff–Equivalence of IMGs and GIRGs

In the following, we concentrate on Pareto–perfect equilibria (PPEs) of IMGs, and on PPEs in simple

paths in GIRGs. The set of potential endpoints in IMGs is calledE, and the endpoints that can still

be reached in a subgame following a stagek with st = s is Es = {e∈ E : e≥ s}. In PPEs of IMGs,

the continuation equilibrium of such a subgame must imply one of the Pareto–efficient endpoints in

Es. Moreover, the set of potential focus points in GIRGs isF , and the focus points that can still be

reached in a stage withsmin(k) = s is Fs = { f ∈ F : f ≥ s}.

The set of endpointsE in IMGs is unique. Consider a PPE of an GIRG ˆs for which there is a

stagek such that (usings = smin(k)) the resulting focus point is not a Pareto efficient point inEs.

Define the setS′ ⊆ S that contains thesmin(k) for all k where such violations occur. We concentrate

on a stagek implying smin(k)∈maxS′, and there it holds for all following stagesk′ that if smin(k′) > s

then the induced focus point is a Pareto efficient point inFsmin(k′) ≡ Esmin(k′). Let f denote the focus

point implied in the continuation equilibrium followingk and letsr denote the restrictions that the

players eventually restrict to along the equilibrium path followingk.

First, consider the casesr 6= s, i.e. the restrictions are adapted in the subgame followingk. By

assumption, the focus pointf is a Pareto efficient point inEsr but not a Pareto efficient point inEs.

Since all points that are not Pareto efficient inEs are Pareto dominated by at least one of the Pareto

efficient points inEs, the path implyingf must not be Pareto perfect.

Secondly, consider the casesr = s, i.e. the restrictions are not adapted, but a focus pointf results

which is not a Pareto efficient point inEs. Here, let us distinguish two subcases:s would result in

a corresponding IMG subgame,s∈ Es, or it would not,s /∈ Es. If s /∈ Es, then for any pointf ≥ s

such thatf is not Pareto efficient inEs, there is an endpointe∈ Es and a playerb who is better off

in e than in f (by construction ofE). Assume that playerb deviates unilaterally from ˆsby restricting

himself to play strategiessb ≥ eb. Thanks to Lemma 3.13, we know that all endpoints, which are

Pareto efficient inEs and more competitive than(eb,s−b) must provideb with a payoff that is at least

as high as his payoff ine. By assumption, the path leads from(eb,s−b) to one of these points, and

hence,b gains by deviating unilaterally from ˆs. Thus,s is not an SPE.

Alternatively, if s∈ Es (i.e. the players would coordinate ons in the IMG), thens is the unique

Pareto efficient endpoint inEs. By assumption,f 6= s. Construct the setE′s = {e∈ E : e> s} as

the endpoints that are strictly more competitive thans. If f is not inE′s, then there is a player who

would profitably deviate to someeb (as in the previous case), and ˆs is not an SPE. Iff is in E′s, then

the play of f is Pareto dominated bys. Moreover, no player has incentives to deviate unilaterally

from restrictions tos, as the resulting endpoint must be Pareto dominated. As a result, sticking to

restrictions ofs and to the focus pointf = s is the unique Pareto efficient equilibrium. In turn, the

abovely assumed profile ˆs is not renegotiation proof. Hence, the GIRG focus points that can result

are equivalent to the IMG endpoints that can result.

21

Proposition 4.2 Consider an arbitrary simple–path PPE ˆs′ of a GIRG and denote the set of potential

focus pointsF . Consider an arbitrary PPE ˆs′′ of an IMG and denote the set of potential endpoints

E. E andF are equivalent for all constituent games that are competitive. Moreover, for anys∈ S,

consider a GIRG stagek′ such thatsmin(k′) = s and an IMG stagek′′ such thatst = s. Both, the

focus point f (k) induced in ˆs′ and the endpointe(k) induced in ˆs′′, are Pareto efficient points in

Es≡ Fs = { f ∈ F : f ≥ s}, and for all such points, there are corresponding equilibria.

Implicitly, the set of focus pointsF induced in PPEs of GIRGs is unique, one of the points in

minF results along the equilibrium path, and there is a unique focus point that can result along an

equilibrium path which Pareto dominates (weakly) the stage game Nash equilibriums∗.

4.4 Perturbed Infinitely Repeated Games

PIRGs are obtained when we adapt generalized repeated games by introducing a move of Nature in

each stage, and this move determines whether the players are allowed to lock in their strategies in the

current stage (nt = 1 with probabilityπ > 0) or not (nt = 0 with probability 1−π > 0). The players

are informed about Nature’s move before they decide. Ifnt = 1, the players’ flags have to satisfy

ft,b ∈ {0,1}, and ifnt = 0 the flags have to satisfyft,b ∈ {0}. The set of stages is

K ={(n0,s0, f0,n1,s1, f1, . . . ,nt ,st , ft) : t ≥ 0, s0 = minS,

∀t ′ ≤ t : (nt ′,st ′, ft ′) ∈ {0,1}×S×{0,1}2, ft ′,b ≤ nt ′ ∀b,

∀t ′ ≤ t : st ′ ≥ smin(s0, f0, . . . ,st ′−1, ft ′−1)}. (25)

The restrictionssminb (k) and the pathp(k) apply as they are define above. A strategy ˆsb maps all

stages to feasible stage strategies. Again, we concentrate on PPEs insimple paths, but we shall adapt

this restriction to acknowledge that the focus points may depend on how Nature moves. Ex post, the

moves of Nature following a stagek can be described as a sequencen∞ ∈ {0,1}∞, and we require

that the focus point is unique only ex post: for each stagek∈ K and for each forthcoming sequence

n∞ of moves of Nature, there is a unique stage strategy profile being played infinitely often along

the equilibrium path (thefocus point). Since the moves of nature are unknown ex ante, the focus

points that may result need to satisfy certain consistency requirements, but those are not relevant in

the following analysis (hence, we skip formalizations of them).

Focus points that result only with probability zero are payoff–irrelevant and can therefore be ne-

glected when we show that the equilibria of PIRGs are payoff–equivalent to the equilibria of IMGs.

Let f (k) denote the set of focus points that can result with positive probability in the subgame fol-

lowing stagek∈ K, and letF denote the set of focus points that can result with positive probability

in arbitrary subgames.

We want to show thatF is equivalent to the set of potential IMG endpointsE. In the arguments

leading to Proposition 4.2, we showed that players would deviate from focus points that can not result

22

in corresponding subgames of IMGs, and we relied on at most one round where the players adapted

their commitments. In any stagek of a PIRG, the players will have an opportunity to do so almost

surely (regardless of how smallπ is). Hence, the corresponding IMG endpoints would result almost

surely, and deviations from IMG endpoints would not occur with positive probability. This argument

leads to the following proposition.

Proposition 4.3 Consider a PPE in simple paths of a PIRG with an arbitraryπ > 0 inducing the

potential focus pointsF , and an IMG PPE inducing the set of endpointsE. F andE are equivalent

for all constituent games that are competitive. Moreover, consider an arbitrary PIRG stagek inducing

restrictionss= smin(k), and denote the set of focus points that are sustained with positive probability

in the subgame following stagek as f (k). The set f (k) is a subset of the focus points that can

result in corresponding IMG subgames,f (k) ⊆ EPs with EP

s as the set of Pareto efficient points in

Es = {e∈ E : e≥ s}.

Apparently, forπ approaching zero, the PIRGs approach IRGs. Hence, marginal perturbations

of infinitely repeated games suffice to reduce the set of payoff allocations that can be sustained

in simple–path PPEs to the set of payoff allocations that can be sustained in IMGs. This is quite

remarkable, as (for constituent games with large strategy sets) the assumption of simple paths is

hardly restricting. Moreover, since marginal perturbations of IRGs suffice to generate this effect,

our approach is merely refining the equilibria of IRGs. This is interesting, as the usual refinement

concepts are rarely effective in IRGs (see, e.g. Rubinstein, 1980).

5 Discussion

Above, we have analyzed strategic irreversibilities, and mainly, we did so with respect to the unique-

ness of collusive equilibria. Judged by our results, deterrence and tacit collusion is to be observed

frequently (i.e. nearly as frequently as irreversible moves). These results are supported by some

experiments on multiple–round auctions with perfect irreversibilities (Breitmoser, 2004a). Theo-

retically, the main exception appears to be the case of strategic substitutes (Cournot competition),

where there may be empty sets of collusive endpoints when it pays to deviate unilaterally beyond

the Nash equilibrium strategy. Empirically, the evidence supporting our theoretical results appears

mixed. Some contradictions might be due to the weakness of the existent collusive endpoints (when

unilateral deviations are cheap), as in the case of Cournot competition. Other contradictions might be

due to imperfections of the irreversibilities, which is more likely to apply to Bertrand competitions

(and it might be worst when the number of competitors is high). An extreme example highlighting

the latter point is reported by Bolle and Baier (2004), who analyze the German call–by–call market.

More contradictions appear to be explainable through referring to emotions (as hate, which may

override rationality), or through referring to inconsistent and/or false beliefs. For instance, consider

23

the situation in Israel. Both sides, the Israel government and Hamas, appear to believe that rigorous

actions bring the other side to give in. In cases like those, where our predictions appear to be falsified,

the introduced model might still be useful to as a basis in illustrations of possible reasons that lead

to escalations of conflicts. In the case mentioned, the contradictions might partly be handled through

assuming bounded rationality on (at least) one of the sides. A still wider range of phenomena might

be explained through altering the assumptions in one of the following directions.

1. static interpersonal concerns (altruism, spite) or dynamical models of emotions (reciprocity),

2. auction–like competitions where overbidding is required,

3. endogenous discretization of the strategy sets, or

4. number of players greater than two.

The first two points are combined in (e.g.) Breitmoser (2004b), where the equilibria are predomi-

nantly collusive as well. In general, however, the introduction of emotions would notably affect the

resulting set of endpoints. The most drastic of effects of interpersonal concerns can be observed in

pure Cournot games (symmetric competition withα = −1, see Section 2.3), when the competitors

aremarginallyspiteful: all collusive endpoints will disappear. Obviously, interpersonal concerns (as

spite or altruism) are likely to be relevant in personal relationships and political games. However,

such concerns are not significantly less appealing in market games. “Altruism” can be thought of as

applying between firms which own shares of each other, or of firms which partly belong to the same

owner (Bolle and Güth, 1992). “Spite” may capture a heuristic resulting from a long–term strategy

aimed at weakening a threatening rival.

The third point concerns the step sizes along which the ideally continuous strategy spaces are

partitioned by the players. Generally, this partitioning affects the outcomes that can be sustained in

renegotiation–proof equilibria, and thus it might be considered to be a part of the game. However, our

results are robust with respect to arbitrary symmetric partitions of the strategy spaces, and asymmetric

partitionings may appear difficult to motivate. In particular, it appears difficult to motivate how the

players can credibly commit to such partitionings. A similar problem arises in irreversible move

games based on non–competitive games. There, it needs be discussed for each class of constituent

games, how the players can credibly restrict their options.

Finally, the above list suggests to extend the analysis to models with more than two players.

Generally, the analysis is likely to become considerably more complex, as a variety of asymmetric

equilibria can result. For example, in public good games partial free riding equilibria are likely to

result.

24

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A Proofs

Lemma A.1 The model of symmetric competition based on (1) satisfies (17).

PROOF. The inverse best reply function is

R−1−b(pk

−b) =1α

(2pk

−b−1)

=1α

(2

2−α+2kε−1

)=

11−α

+2kεα

. (26)

The payoff from sticking to the path(pkb, pk

−b) is

Πb(pkb, pk

−b) =(

1− 12−α

−kε+α

2−α+αkε

)∗(

12−α

+kε)

(27)

=1

(2−α)2 +αkε− (kε)2∗ (1−α). (28)

The payoff from the deviation onto the opponents best reply curve is

Πb

(R−1−b(pk

−b), pk−b

)=(

1− 12−α

− 2α

kε+α

2−α+αkε

)∗(

12−α

+2α

kε)

(29)

=1

(2−α)2 +αkε− (kε)2∗(

4α2 −2

). (30)

Thus, the former is greater than the latter, iff

1−α <4

α2 −2 ⇔ α2(3−α) < 4, (31)

which is satisfied for allα ∈ (−1,0). QED

Lemma A.2 In the model of competition described in (4), playerb is better off deviating unilaterally

from (pkb, pk

−b) to R−1−b(pk

−b) in the stage game if and only ifαb < 0 andαb +α2−b > 0.

PROOF. Thepkb andpk

−b satisfy

pkb := pN

b +kK

(pM

b − pNb

)=

2+αb

4−αbα−b+

kK∗

2α−b +α2b

(4−αbα−b) (2−αb−α−b), (32)

pk−b := pN

−b +kK

(pM−b− pN

−b

)=

2+α−b

4−αbα−b+

kK∗

2αb +α2−b

(4−αbα−b) (2−αb−α−b). (33)

27

UsingR−1−b(pM

−b) = αb+α−bα−b(2−αb−α−b)

, we can calculateR−1−b(pk

−b) as

R−1−b(pk

−b) = pNb +

kK

[R−1−b(pM

−b)− pNb

]=

2+αb

4−αbα−b+

kK∗

4αb +2α2−b

α−b(4−αbα−b)(2−αb−α−b).

(34)

In cases where no playerb is better off deviating, the players collude at(pMb , pM

−b). The payoff from

not deviating is

Πb(pkb, pk

−b) = (1− pkb +αbpk

−b)∗ pkb (35)

=(

2+αb

4−αbα−b

)2

+kK∗ 2+αb

4−αbα−b∗

2α2b +αbα2

−b

(4−αbα−b) (2−αb−α−b)

+(

kK

)2

∗α4

b +α3bα2

−b +2αbα3−b−4α2

−b

(4−αbα−b)2 (2−αb−α−b)2

and the payoff from deviating is

Πb

[R−1−b(pk

−b), pk−b

]=[1−R−1

−b(pk−b)+αbpk

−b

]∗R−1

−b(pk−b)

=(

2+αb

4−αbα−b

)2

+kK∗ 2+αb

4−αbα−b∗

2α2b +αbα2

−b

(4−αbα−b) (2−αb−α−b)

+(

kK

)2

∗8α3

bα−b +8α2bα3

−b−16α2b−16αbα2

−b +2αbα5−b−4α4

−b

α2−b (4−αbα−b)2 (2−αb−α−b)2

Playerb is not better off deviating iff

Πb(pkb, pk

−b)≥Πb

[R−1−b(pk

−b), pk−b

](36)

α4bα2

−b +α3bα4

−b ≥ 8α2bα3

−b−16α2b−16αbα2

−b +8α3bα−b (37)

α3bα2

−b∗(αb +α2

−b

)≥(8α2

bα−b−16αb)∗(αb +α2

−b

)(38)

Obviously, this is fulfilled ifαb + α2−b = 0. Additionally, if αb > 0 or αb + α2

−b < 0 the relation

simplifies to

α2bα2

−b +8(1−αbα−b)︸ ︷︷ ︸>0

≥−8, (39)

which is fulfilled, too. However, if (and only if)αb + α2−b > 0, playerb is better off deviating to

R−1−b(pk

−b), as the relationα2bα2

−b + 8(1−αbα−b) ≤ −8 can not be fulfilled in our circumstances.

QED

Lemma A.3 In the contest according to (8), Equation (17) is satisfied.

PROOF. If it is less more profitable for some playerA to deviate from a symmetric collusive point

(x,x) to the endpoint(R−1−b(x),x), then (in a quasi–continuous case)

x2x−x >

yy+x

−y, with y =

(12

+

√14−x

)2

. (40)

28

which simplifies to

12−x > x ⇔ x <

14

(41)

which is fulfilled for thex in question. QED

Lemma A.4 In a contest according to (9), Equation (17) is satisfied.

PROOF. Equation (17) is equivalent to

Ub

(ln

cPb,ln

cP−b

)≥Ub

(ln

cPb +n− l

nPb,

ln

cP−b

)≥Ub

(R−1−b

(ln

cP−b

),

ln

cP−b

)Pb

Pb +P−b+

n− ln

Pb ≥(n+ lc− l)Pb

(n+ lc− l)Pb + lcP−b

P−b +1Pb

− nlPb ≤

lcP−b

(n+ lc− l)Pb⇔ n− l

n+ lc− l≤ n

l

P2b

P−b+

1P−b

which is satisfied forl ≤ n andP2b +1≤ P−b. QED

Proof of Lemma 3.7. If any playerb deviates from a points= (sb,s−b) ≮ s∗ with sb ≥ Rb(s−b) to

a more competitive strategys′b, then whatever endpoints′′ > s is reached finally,b is worse off since

Ub(s) > Ub(s′b,s−b)≥Ub(s′′). (42)

The first inequality stems fromsb ≥ Rb(s−b) and 3.1.(ii), the second one from 3.1.(i). Hence, any

s : s≥ R(s) is an endpoint. Similarly, we see thats∗ Pareto dominates any points′ > s∗: for each

playerb, he is better off in(s′b,s∗−b) than ins′ and he is better off ins∗ than in(s′b,s

∗−b).

To show thats≥ R(s) is even necessary fors ≮ s∗ being an endpoint, let us assume thatb’s

opponent has moved beyond his Nash strategy,s−b > s∗−b, and defines′b := Rb(s−b). Thanks to

3.1.(iii), s−b > s∗−b implies s−b ≥ R−b [Rb(s−b)] and thus we have(s′b,s−b) ≥ R(s′b,s−b). Hence,

(s′b,s−b) is an endpoint.

If s≥ s∗, there must be someb : sb < Rb(s−b) such thats−b > s∗b. In that case,b can deviate

profitably and unilaterally to the endpoint[Rb(s−b),s−b]. If s� s∗, then there is a playerb : sb < s∗b.

As a result of 3.1.(iii), we can rule out thatsb ≥ Rb(s−b) ands−b < R−b(sb). Hence,sb < Rb(s−b),

andb can deviate unilaterally and profitably to the endpoint[Rb(s−b),s−b]. Thus,s : s� s∗, s� R(s)

can not be an endpoint. QED

Proof of Lemma 3.11. Here, we make use of the notations1∨s2 :=(max{s1,A,s2,A},max{s1,B,s2,B}).The proof results from the following three claims.

Claim A.5 If for any pair of endpointse1 6= e2 < s∗ in an IMG bothe1 � e2 ande2 � e1 hold, then

the pointe3 = e1∨e2 must be outside the interiorSI of the strategy set (proved in the appendix).

29

PROOF OFCLAIM . Suppose to the contrary that there would be two endpointse1,e2 ∈ E that have

e1 � e2, e1 � e2, ande1∨e2 ∈ SI . This implies thate1,e2 ≤ s∗. Without loss of generality, we shall

assumee1,A > e2,A. Sincee3 ∈ SI and thuse3 ≤ s∗, we know

UA(e3) > UA(e1), UB(e3) > UB(e2). (43)

As a result of that,A’s payoff in e3 must be high enough to Pareto dominate all endpointse′ ≥ e1,

which is a superset of all endpointse′ ≥ e3. Similarly,B’s payoff ins3 must be high enough to Pareto

dominate all endpointse′ ≥ e2, which again includes all endpointse′ ≥ e3. Hence,e3 must be an

endpoint itself; and in turn,e1,e2 must not be, since eitherA or B is better off deviating toe3. a

Claim A.6 If for any pair of endpointse1 6= e2 < s∗ both e1 � e2 ande2 � e1 hold, then the point

e3 = e1∨e2 must be an endpoint, too.

PROOF OFCLAIM . Supposee3 < s∗ is not an endpoint. Let us definee′ as an arbitrary endpoint that

is more competitive thane3, i.e.e′ ∈ {s∈ E : s> e3}. If e3 is outside the interior, there is a playerb

such thate3,b > Rb(e3,−b). Since the payoff functions are unimodal and monotonically decreasing in

the opponent’s competitiveness, this implies

Ub(e3) > Ub(e′b,e3,−b) > U(e′) (44)

that b is strictly better off ine3 than in any more competitive endpoint. Thanks to characteristic

3.1.(iii), we know thate3,−b < R−b(e3,b). Without loss of generality, assume thate1,−b < e3,−b.

Hence,b’s opponent is better off ine3 than ine1, and sincee1 is an endpoint, he is better off ine1

than in all more competitive endpoints. This extends toe3, and thus, both players are better off ine3

than in any more competitive endpoint. As a result,e3 must be an endpoint ande1 must not be an

endpoint. a

Claim A.7 There are no endpointse< s∗ that are outside the interior.

PROOF OFCLAIM . Suppose to the contrary,e < s∗ is an endpoint that is outside the interior and

define playerb such thateb > Rb(e−b). Now, constructe′−b = R−1b (eb). Equivalently to argument

leading to (44),b is better off in(eb,e′−b) than in any more competitive point. Moreover, in competi-

tive games,e′−b < R−b(eb), and thusb’s opponent is better off in(eb,e′−b) than ine. Thus, similarly

to the above arguments,(eb,e′−b) is an endpoint andemust not be an endpoint. a

Combined, these claims complete the proof. QED

Proof of Lemma 3.12. Only a point in minE can result along the equilibrium path. Sinces∗ ∈ E,

there must be a pointe∈minE such thate≤ s∗. Moreover, as a result of Lemma 3.11, the endpoint

30

e∈minE : e≤ s∗ is unique. That is, there is a unique endpointe≤ s∗ that can result in equilibrium,

and there is generally such an endpoint.

Alternatively, some endpointse≮ s∗ may result in equilibrium. In all of these endpoints, how-

ever, at least one of the players is worse off than ins∗. For, in all of these endpointse there is some

playerb : eb < s∗b, and this player is better off ins∗ than ine. Precisely, he is better off(eb,s∗−b) than

in e, and better off ins∗ than in(eb,s∗−b). Hence, the unique endpoint that can result in equilibrium

where no player is worse off than ins∗ is e∈minE : e≤ s∗. QED

Proof of Lemma 3.13. SupposeEs is the set of potential endpoints in a stagek ∈ KD with st = s,

andEPs is the set of Pareto efficient points inEs, which impliesEP

s ∈ minEs. As a result, for each

pair of pointse′ 6= e′′ ∈ EPs , we have (for all playersb) thate′b > e′′b ⇔ e′−b < e′−b.

There are three groups of potential endpoints that we have to establish the claimed payoff relation

for. First, endpointsemay satisfyeb < s∗b ande−b > s∗−b; secondly, they may satisfye≤ s∗ (of which

there is only one such point); thirdly, the may satisfyeb > s∗b ande−b < s∗−b. We have to proof the

claim for all possible combinations of these groups: two points within group 1, two points within

group 3, group 1 vs. group 2, group 2 vs. group 3, and group 1 vs. group 3.

First, we relate a pointe1 of group 1 tos∗. Sincee1−b > s∗−b, we haveUb(e1) < Ub(e1

b,s∗−b).

Moreover,e1b < Rb(s∗−b) impliesUb(s∗) > Ub(e1

b,s∗−b), and hence,Ub(s∗) > Ub(e1). The above re-

lations hold equivalently, when we substitutes∗ with an arbitrary pointe′ of group 1 that satisfies

e′b > e1b, or when we substitutes∗ with an arbitrary pointe2 < s∗ of group 2. Thus, for the first two

combinations of groups, the claim is proved.

Secondly, we relate a pointe3 of group 3 tos∗. Equivalently to the above induction, we can

show that the payoff ofb’s opponent is higher ins∗ than ine3, i.e.U−b(s∗) > U−b(e3). However,

e3 is known to be Pareto efficient, which can only be satisfied ifUb(s∗) < Ub(e3). Equivalently, this

relation holds if we substitutes∗ with an arbitrary pointe′ of group 3 that satisfiese′b < e3b, or when

we substitutes∗ with an arbitrary pointe2 < s∗ of group 2.

Finally, we have to relate pointse1 of group 1 to pointse3 of group 3. In any subgame with

restrictions, where both kinds of endpoints can be reached, we haves� s∗. Thus, there is an interior

endpoint (at leasts∗) that can be reached, too, and the endpointse1 ande3 are not Pareto dominated

by that one. Thus, we can combine the above cases to prove the claimed payoff relation: playerb is

better off ins∗ than ine1, and he is better off ine3 than ins∗. Hence, he is better off ine3 than ine1.

QED

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