D O C U M E N T O D E T R A B A J O

Instituto de EconomíaD

OC

UM

ENTO

de TR

ABA

JOI N S T I T U T O D E E C O N O M Í A

www.economia.puc.cl • ISSN (edición impresa) 0716-7334 • ISSN (edición electrónica) 0717-7593

A Dominance Solvable Global Game with Strategic Substitutes

Rodrigo Harrison; Pedro Jara-Moroni.

4402013

1

A Dominance Solvable Global Game with Strategic

Substitutes

⇤

Rodrigo Harrison † Pedro Jara-Moroni ‡

Abstract

Global games emerged as an approach to equilibrium selection. For a general settingwith supermodular payo↵s unique selection of equilibrium has been obtained throughiterative elimination of strictly dominated strategies. For the case of global gameswith strategic substitutes, uniqueness of equilibrium has not been proved by iterativeelimination of strictly dominated strategies, making the equilibrium less appealing.In this work we study a simple three player binary action global game with strategicsubstitutes for which we provide a condition for dominance solvability. This opens anunexplored research agenda on the study of global games with strategic substitutes.

1 Introduction

Global games are games of incomplete information, where the players’ payo↵s depend onan uncertain state that represents the fundamental of the modeled situation, from whicheach player receives a private signal with a small amount of noise. In these games, the noisetechnology is common knowledge so each player’s signal generates beliefs about fundamentalsof the model and the other players’ beliefs (over fundamentals and beliefs of their rivalsand so on). Incomplete information comes from a noisy payo↵ perturbation of a completeinformation game in a way that when the noise vanishes we recover the original game.Originally, global games were assessed as equilibrium selection devices and, in time, theyhave become as well a useful methodology to simplify the analysis of high-order beliefs instrategic settings. Our interest relates to their equilibrium selection application.

Global games were first introduced by Carlsson and van Damme (1993) as a means todepart from the assumption that players are excessively rational and well-informed withrespect to the real-life situation under scrutiny. The idea behind this equilibrium selection

⇤We thank Stephen Morris for valuable conversation and insights and the audiences at UECE LisbonMeetings 2011, 2012 Paris Conference of the International Network on Expectational Cordination, the fourthWorld Congress of the Game Theory Society, the 2013 meeting of the Chilean Society of Economics, and atseminars at Universidad Catolica de Chile and Universidad Adolfo Ibanez for their comments.

†harrison@uc.cl, Instituto de Economıa, Pontificia Universidad Catolica de Chile.‡pedro.jara@usach.cl, Departamento de Economıa, Universidad de Santiago de Chile.

1

approach is to examine the set of Nash equilibria of a game as a limit of equilibria ofpayo↵-perturbed games and observe any reduction in the set. For a given realization of thestate and its associated complete information game, the global game approach may allowselection of a unique equilibrium in this game, provided that there is a unique equilibriumin the incomplete information game that results when the noise in the players’ observationis su�ciently small.

Carlsson and van Damme (1993) show that for a general class of two-player, two-actiongames, this limit comprises a single equilibrium profile. Moreover, the equilibrium profile isobtained through iterative elimination of strictly dominated strategies (henceforth IESDS).Roughly, the deletion requires that, for each player and for each action of that player, thereare certain extreme values of the state, for which that action is strictly dominant. Even ifthese values carry very little probability weight, the players can use signals close to these“dominance regions” to rule out certain types of behavior of others. Hence, the iterativedeletion proceeds.

These results have been extended by Frankel et al. (2003) to a more general class of globalgames with strategic complementarities which have been useful for the study of economicmodels such as bank runs, currency crises and herding behavior, among others.1

Global games with strategic substitutes had not been as thoroughly studied as the caseof strategic complements. Uniqueness of equilibrium can not be obtained by simply passingfrom the strategic complements model of Frankel et al. (2003) to the strategic substitutesenvironment. The elimination of strictly dominated strategies may not provide a uniqueoutcome and so this technique cannot be used to prove uniqueness of equilibrium. However,by adding a minimum of player heterogeneity, Harrison (2003) showed that the equilibriumis unique in a fairly general model with strategic substitutes. Still, this unique equilibriummay not be the unique outcome of the IESDS (Morris, 2009). In accord with the literature onstrong rationality the predictive power of the global game approach for equilibrium selectioncomes not only from uniqueness of equilibrium but also from the method by which thisequilibrium is obtained. This is one of the reasons why we are interested in dominancesolvability.2

In the light of dominance solvability results in games with strategic substitutes and com-plete information (Zimper, 2007; Guesnerie and Jara-Moroni, 2011), further requirementsshould allow stating that this unique equilibrium is in fact the only strategy profile that sur-vives IESDS. In this article we study a simple three player global game with strategic substi-tutes with heterogenous players that satisfies the conditions for uniqueness of equilibrium ofthe theorem in Harrison (2003).3 We show that if players are su�ciently heterogeneous, theprocess of IESDS delivers the unique equilibrium profile. This result solves a puzzle in the

1For a complete survey of the global games literature see Morris and Shin (2003)2The concept of strongly rational equilibrium was first stated by Guesnerie (1992) as a mean to provide

an eductive foundation for the rational expectations hypothesis. An equilibrium is strongly rational, if itis the only rationalizable strategy profile of a game (Guesnerie, 1992, 2002). Dominance solvability impliesstrong rationality of the equilibrium.

3The simplest unexplored case is the three player binary action game, since results in two player globalgames may be derived from Carlsson and van Damme (1993).

2

global games literature and resembles the results found in Zimper (2007) and Guesnerie andJara-Moroni (2011), regarding the passage from strategic complements to strategic substi-tutes. It is indeed possible to obtain dominance solvability under strategic substitutes, butuniqueness of equilibrium is not su�cient as in the case of strategic complements. Additionalconditions must be required.

2 A simple global game with strategic substitutes

In this section we introduce the global game under scrutiny. We present a binary actionthree player game with strategic substitutes and heterogeneous players.

2.1 Setting

Consider a three player binary action game characterized by the payo↵s ui

: {0, 1}3⇥R! R,4

ui

(ai

, a�i

, x) := ai

✓d

2(3� a

1

� a2

� a3

) +mx� ci

◆for i 2 {1, 2, 3}

where m > 0, d > 0 represents the degree of strategic substitution5 and ci

may be interpretedas player i’s specific cost. Heterogeneity of players is introduced by assuming that 0 < c

1

<c2

< c3

.

Note that player i’s payo↵ function is of the form ui

(ai

, a�i

, x) = ⇡i

⇣ai

,P

j 6=i

aj

, x⌘, where

⇡i

: {0, 1}⇥ {0, 1, 2}⇥R! R is is an auxiliary function defined by

⇡i

(ai

, n, x) := ai

✓d

2(3� a

i

� n) +mx� ci

◆

that depends on other players’ actions through their sum (the number of players -other thani- that are choosing action 1).

Let us define �⇡i

(n, x) = ⇡i

(1, n, x) � ⇡i

(0, n, x) as the net gain of player i of playing 1instead of 0. Then

�⇡i

(n, x) = �⇡(n, x)� ci

where

�⇡(n, x) :=d

2(2� n) +mx.

Note that in this model

�⇡i

(n, x)��⇡i

(n+ 1, x) = d (1)

4This game is inspired by the game presented in Morris (2009).5See equation (1).

3

and since d > 0, the incentive to choose the higher action is decreasing in the actions of therivals, so the game indeed presents strategic substitutes.6 The greater the value of d, thesteeper the incentive to play the higher action.

Finally, let us define ki

(n) by the unique solution in x of

�⇡i

(n, x) = 0 =) ki

(n) =ci

� d

2

(2� n)

m. (2)

The value ki

(n) allows us to identify the optimal action of player i when n of her opponentsare playing 1. If x < k

i

(n) then player i will choose action 0 and if x > ki

(n) then player iwill choose action 1. Note that k

i

(n) is increasing in n. We will denote

ki

= ki

(0) ki

= ki

(1) ki

= ki

(2)

We see then that if x > ki

then player i will optimally choose action 1 regardless of thenumber of opponents playing 1. Equivalently, if x < k

i

then player i will optimally chooseaction 0 regardless of the number of opponents playing 1. In the global game literature,these intervals are called upper and lower dominance regions, respectively.

If there is complete information, depending on the value of x the game may have: mul-tiple equilibria, a unique equilibrium, a unique equilibrium with two players playing strictlydominant strategies or a unique equilibrium in strictly dominant strategies (when x is in thedominance regions of all the players). Figure 1 depicts the type of equilibria depending onthe value of x and the dominance regions for each player.

Since we have multiplicity of equilibria for some values of x, we are interested in usingthe global game approach for equilibrium selection. However, we are not only interested inuniqueness of equilibrium under incomplete information but also in the possibility that thisprofile is obtained through IESDS.

x

Player i’s action

0

1

k1 k2 k3 k1 k2 k3 ¯

k1¯

k2¯

k3

Player 1

Player 2

Player 3

(0, 0, 0) (1, 0, 0) multiplicity (1, 0, 0) (1, 1, 0) multiplicity (1, 1, 0) (1, 1, 1)

Figure 1: Dominance regions and equilibria.

6A negative d would model a game with strategic complements.

4

2.2 Incomplete Information

Consider now the three player incomplete information game �(�), consisting of the previouspayo↵ structure and where each player has some uncertainty about x. Instead of observingthe actual value of x, each player just observes a private signal x

i

, which contains di↵useinformation about x, which is composed of the true value plus some noise:

xi

= x+ �"i

where � > 0 is a scale factor, x is drawn from a su�ciently large7 interval [X,X] withuniform density and each "

i

is randomly selected independently of x on the interval⇥�1

2

, 12

⇤.

In this context, each signal xi

belongs to the set X(�) = [X � 1

2

�, X + 1

2

�] and players careabout the distribution of their rivals’ signals. This general noise structure has been used inthe global game literature, allowing the conditional distribution of each opponent’s signalto be modeled in a simple way, i.e. given a player’s own signal, the conditional distributionof an opponent’s signal x

j

admits a continuous density f�

and a cdf F�

with support in theinterval [x

i

� �, xi

+ �]. Moreover this literature establishes a significant result; when theprior is uniform, players’ posterior beliefs about the di↵erence between their own observationand other players’ observations are the same, i.e. F

�

(xi

| xj

) = 1� F�

(xj

| xi

).8

If "j

� "i

is distributed according to H, then the probability that player j’s signal xj

issmaller than a value t given the signal x

i

received by player i, Pr(xj

t|xi

) = F�

(t|xi

), iscalculated using H as follows:

F�

(t|xi

) = Pr(xj

t|xi

) = Pr⇣"j

� "i

(t�xi)

�

⌘= H

�t�xi�

�.

In this note we assume that H is the cdf function of the uniform distribution on [�1, 1].A Bayesian pure strategy for a player i is a function s

i

: X(�) ! {0, 1}. A pure strategyprofile is denoted as s = (s

1

, s2

, s3

) and si

2 Si

, the set of all functions from X(�) to {0, 1}.A switching strategy of player i is a Bayesian pure strategy s

i

satisfying:

9 y s.t. si

(xi

) =

(0 if x

i

< y

1 if xi

> y(3)

Abusing notation, we write si

(·; y) to denote the switching strategy of player i with switchingthreshold y.

Consider the following strategy profile s⇤:

s⇤1

= s1

(·; k1

) s⇤2

= s2

(·; k2

) s⇤3

= s3

�·; k

3

�(4)

Following Harrison (2003), if � is su�ciently small the strategy profile s⇤ is the unique BNEof �(�).9 This result does not say anything about dominance solvability. The proof in

7Such that ki(n) 2⇤X,X

⇥for all i and all n.

8This property holds approximately when x is not distributed with uniform density but � is small, i.e.F (xi | xj) ⇡ 1� F (xj | xi) as � goes to zero. See details in Lemma 4.1 Carlsson and van Damme (1993).

9The reader may check that this payo↵ structure satisfies assumptions (A1) through (A5) of Harrison(2003).

5

x

Player i’s action

0

1

k1 k2 k3 k1 k2 k3 ¯

k1¯

k2¯

k3

Player 1

Player 2

Player 3

Figure 2: Equilibrium in the global game.

Harrison (2003) utilizes a much stronger concept than IESDS. Moreover, it is possible togive examples where the unique equilibrium is not strongly rational (Morris, 2009). In thenext section we present a su�cient condition under which �(�) is dominance solvable.

3 Main Result

Dominance regions play a central role in the process of equilibrium selection in global games.In general, global games with strategic complements require the existence of these regions(plus other assumptions related with continuity and monotonicity of the payo↵ functions)in order to do the equilibrium selection through iterative elimination of strategies. However,following Harrison (2003), we realize that, in the strategic substitute case, in order to startany process of elimination we need, additionally to the existence of dominance regions, someplayer’s heterogeneity. Nevertheless, this condition is not su�cient for dominance solvability.In accord with the global game literature, we study the process of IESDS and identify asu�cient condition under which this procedure delivers a unique profile in �(�). Thus, wefind a condition under which the standard global games results (Carlsson and van Damme,1993; Frankel et al., 2003) can be extended to this environment of strategic substitutes.

Our main result is summarized in the following proposition.

Proposition 1. If

c3

� c1

>d

2

then 9 � > 0 such that 8 � < �, the profile s⇤ defined in (4) is the unique profile surviving

iterative elimination of strictly dominated strategies.

Proposition 1 states that for a given degree of substitution (d) if players are su�cientlyheterogenous, c

3

� c1

> d

2

, or equivalently, if given players’ heterogeneity the degree ofstrategic substitution is su�ciently small, then we get dominance solvability in �(�) for a

6

su�ciently small �. Furthermore, it implies that the unique equilibrium of �(�) is stronglyrational, in the terminology of Guesnerie (1992) (see footnote 2), contributing to the litera-ture of Expectational Coordination.

The approach of the proof, which is relegated to the appendix, is based on the standardmethod used in the global game literature. Mainly, we study a process of elimination ofstrategies that, in each step, finds it to be unreasonable for players to use one of the twoactions when their signal is below or above certain values of the parameter x. These valuesare updated in each step and so constitute two sequences for each player i. The sequence{xt

i

}1t=0

that decreases starting from ki

, is such that at each step t: player i 2 {1, 2} has asa dominant action the action 1 when her signal is above xt

i

and player 3 has as a dominantaction the action 0 when her signal is above xt

3

and below k3

. The sequence {xt

i

}1t=0

thatincreases starting from k

i

, is such that at each step t: player i 2 {2, 3} has as a dominantaction the action 0 when her signal is below xt

i

and player 1 has as a dominant action theaction 1 when her signal is below xt

1

and above k1

. The updating is known as the “contagione↵ect”. We show that under the assumptions of the Proposition, if � is su�ciently small, thenthese sequences cross a threshold equal to the switching point of the equilibrium strategy ofplayer 2, k

2

, thus fixing the only possible remaining strategies at the equilibrium ones.To better understand how this result fits into the previous global game literature, consider

the game described in Section 2 but where the parameter d can also take negative values,allowing the game to be of strategic substitutes or complements depending on the sign ofd. We call this new game �(d, �). If d is negative, then Frankel et al. (2003) show thatwhen � ! 0 the global game �(d, �) has a unique equilibrium obtained through IESDS. Ifd is positive, Harrison (2003) shows that when � ! 0 the equilibrium in �(d, �) is unique.Further, if d is positive and less than 2(c

3

� c1

), Proposition 1 states that when � ! 0this unique equilibrium is the only outcome of IESDS. Then, we may state the followingproposition.

Proposition 2.

• If d < 2 (c3

� c1

), then when � ! 0, �(d, �) is dominance solvable.

• If d � 2 (c3

� c1

), then when � ! 0, �(d, �) has a unique equilibrium.

4 Conclusions

We have presented a simple model of a global game with strategic substitutes for whichwe provide a su�cient condition for dominance solvability, extending previous results inthe literature of global games with strategic complements to this particular case. It is wellknown that expectational coordination is strongly linked to uniqueness of equilibrium understrategic complements, but that uniqueness is not su�cient for expectational coordinationunder strategic substitutes (Milgrom and Roberts, 1990; Guesnerie, 2005). However, furtherrequirements on the primitives of the model may indeed provide predictive power to the

7

unique equilibrium in the world of strategic substitutes and complete information (Zimper,2007; Guesnerie and Jara-Moroni, 2011). Analogously, we have identified a condition thatprovides an optimistic stability result for equilibrium selection in global games under strategicsubstitutes.

The condition highlights a relation between the degree of strategic substitution and het-erogeneity among players. It states that given a degree of strategic substitution playersmust be su�ciently heterogeneous, or equivalently, that given players’ heterogeneity, thereis a maximal degree of strategic substitution.

The intuition behind this condition may be illustrated from the classical results in gameswith complete information. In general dominance solvability is determined by the structureof the game. We know that in games with strategic complements the elimination of strategiesdoes not depend on the degree of complementarity. The elimination process is representedby monotone sequences of strategies that converge to equilibria. Thus, unique equilibriumimplies dominance solvability. Under strategic substitutes, the elimination process is rep-resented by non-monotone sequences of strategies.10 Rather, these sequences are “cyclical”and may not converge to equilibria. Thus, unique equilibrium may not imply dominancesolvability. In our particular case, heterogeneity allows the process to start. Furthermore, asin the case of complete information, a low degree of substitution generates more “drastic”sequences of strategies pushing their limits to be the unique equilibrium.

Our result is consistent with Guesnerie and Jara-Moroni (2011) and Morris and Shin(2009) and suggests that we may have dominance solvability in a more general set up ofglobal games with strategic substitutes. From the condition in Proposition 1 and equation(1) we conjecture that the result may be obtained for more general payo↵s of the form�⇡(n, x) � c

i

that satisfy assumptions (A1) to (A5) in Harrison (2003). This last remarkopens an unexplored research agenda in the study of strongly rational equilibria in globalgames with strategic substitutes.

References

Carlsson, H. and E. van Damme (1993, September). Global games and equilibrium selection.Econometrica 61 (5), 989–1018.

Frankel, D. M., S. Morris, and A. Pauzner (2003, January). Equilibrium selection in globalgames with strategic complementarities. Journal of Economic Theory 108 (1), 1–44.

Guesnerie, R. (1992, December). An exploration on the eductive justifications of the rational-expectations hypothesis. The American Economic Review 82 (5), 1254–1278.

Guesnerie, R. (2002, March). Anchoring economic predictions in common knowledge. Econo-metrica 70 (2), 439.

10It is so in the underlying process of IESDS.

8

Guesnerie, R. (2005, July–August). Strategic substitutabilities versus strategic complemen-tarities: Towards a general theory of expectational coordination? Revue d’Economie

Politique 115 (4), 393–412.

Guesnerie, R. and P. Jara-Moroni (2011). Expectational coordination in simple economiccontexts. Economic Theory 47, 205–246. 10.1007/s00199-010-0556-8.

Harrison, R. (2003). Global games with strategies substitutes. Working paper 03-06, George-town University.

Milgrom, P. and J. Roberts (1990, November). Rationalizability, learning, and equilibriumin games with strategic complementarities. Econometrica 58 (6), 1255–1277.

Morris, S. (2009). Three player global game with strategic substitutes and many rationaliz-able actions. Personal comunication.

Morris, S. and H. S. Shin (2003). Global games: Theory and applications. In M. Dewa-tripont, L. Hansen, and S. Turnovsky (Eds.), Advances in Economics and Econometrics:

Theory and Applications: Eighth World Congress Vol I, Volume 1, pp. 57–114. CambridgeUniversity Press.

Morris, S. and H. S. Shin (2009). Coordinating expectations: Global games with strategiessubstitutes.

Zimper, A. (2007, September). A fixed point characterization of the dominance-solvabilityof lattice games with strategic substitutes. International Journal of Game Theory 36 (1),107–117.

A Appendix

Proof of Proposition 1. Since we have the dominance regions, we know that in any reasonablestrategy player i plays 0 when the signal is below k

i

and plays 1 if the signal is above ki

.Thus, we start the process of elimination of strategy profiles by considering, for each player,strategies of the form:

si

(xi

) =

(0 if x

i

< ki

1 if xi

> ki

.

Between ki

and ki

the strategies may take any value. We make the analysis for the sequencesxt

i

such that, in each step t, strategies take a unique value when xi

is below xt

i

; the analysisfor the sequences xt

i

is analogous. We set then x0

i

= ki

.Since we want to isolate the equilibrium strategy profile, we want to show that for player

1, receiving a signal below xt

1

makes her play 1 (since below k1

her strategy is already fixed at

9

0) and for players 2 and 3 receiving a signal below xt

i

makes them play 0. Since the game isof strategic substitutes, we only need to consider that players observe the worst case scenarioand update the values xt

i

below which their strategies are fixed as above. If this is true forthis scenario, then it is true for any other scenario. The updating occurs as follows: giventhe actual values of the sequences of players 2 and 3, we update the value of the sequence ofplayer 1. With his new value of player 1, we update the values of players 2 and 3. We thenstudy the limits of the sequences xt

i

.Let us call ⇧

i

(ai

, s�i

, xi

) the expected payo↵ of choosing action ai

when player i is observ-ing a signal x

i

and is facing a strategy profile s�i

. Her expected net gain of choosing action1 instead of action 0 can be written as �⇧

i

(s�i

, xi

). In each step t each player i knows theaction prescribed by her opponents’ strategies for values below the thresholds xt�1

�i

. Then,since we want to isolate the equilibrium strategy for each player, for player one we wantto show that she is playing 1 below xt

1

and for players 2 and 3 we want to show that theyare playing action 0 below xt

i

. The worst case scenario then for player 1 is considering thather opponents are playing action 1 whenever there is uncertainty about their play; and forplayers 2 and 3 the worst case scenario is considering that their opponents play action 0whenever there is uncertainty about their play.

At each step t, player 1 knows that any remaining strategy of player i 2 {2, 3} has thefollowing structure:

si

(xi

) =

(0 if x

i

< xt�1

i

1 if xi

> xt�1

i

.

Thus, the worst case scenario for player 1 is such that her opponents are playing

st�1

i

(xi

) =

(0 if x

i

< xt�1

i

1 if xi

> xt�1

i

.

This is, that they are playing a switching strategy with threshold xt�1

i

. We have that�⇧

1

(st�1

2

, st�1

3

, xt�1

1

) > 0 and so we look for a new value xt

1

> xt�1

1

such that�⇧1

(st�1

2

, st�1

3

, xt

1

) =0. The update equation for player 1 is then

0 = �⇧1

(st�1

2

, st�1

3

, xt

1

)

= �⇡�0, xt

1

�F�xt�1

2

| xt

1

�F�xt�1

3

| xt

1

�+

�⇡�1, xt

1

� F�xt�1

2

| xt

1

� �1� F

�xt�1

3

| xt

1

��+�1� F

�xt�1

2

| xt

1

��F�xt�1

3

| xt

1

� �+

�⇡�2, xt

1

� �1� F

�xt�1

2

| xt

1

�� �1� F

�xt�1

3

| xt

1

��� c

1

(5)

where F (xj

| xi

) is the probability that player j observes a signal below xj

given that playeri receives a signal x

i

.Note that the right hand side of equation (5), can be decomposed as a function of y = xt

1

,denoted G

x

t2x

t3(y), minus c

1

. The function Gx

t2x

t3(y) is equal to �⇡(0, y) when y is small, then

10

Figure 3: The expected payo↵ for player 1. The vertical lines are xt�1

2

and xt�1

3

.

it jumps down (because � is small) continuously to �⇡(2, y) when near xt�1

2

and xt�1

3

andthen it is equal to �⇡(2, y) when y is large (see Figure 3). Equation (5) seeks a point suchthat this function is equal to c

1

.We have then either one or three solutions. If there is only one solution, then it is either

k1

or k1

. In the first case we must interpret that xt

1

is 1 and thus the sequence wouldhave collided with xt

1

and we would have already isolated the equilibrium strategy for player1. The second case is not possible under our assumptions (the existence of the dominanceregions). If we have three solutions then xt

1

is in the jump. In this case and since xt�1

2

andxt�1

3

are both in the jump of their function (see below), we must have that at the jump bothF�xt�1

2

| xt

1

�and F

�xt�1

3

| xt

1

�are di↵erent from 0 and 1 and so we may replace them by the

expression for the increasing part of F . This gives:

xt

1

=d�xt�1

2

+ xt�1

3

+ 2��� 4c�

2 (d� 2m�). (6)

Analogously, we can construct the worst case scenarios for players 2 and 3 and determinethe update equations for xt

2

and xt

3

. Recall that the worst case scenario for them is consideringtheir opponents playing 0 whenever there is uncertainty about their play. So the updateequations are:

0 = �⇡�0, xt

2

� �1� F

�xt

1

| xt

2

��+�⇡

�1, xt

2

�F�xt

1

| xt

2

�� c

2

; (7)

0 = �⇡�0, xt

3

� �1� F

�xt

1

| xt

3

��+�⇡

�1, xt

3

�F�xt

1

| xt

3

�� c

3

. (8)

The right hand sides of equations (7) and (8), as functions of each player’s signal have theform G

x

t1(y) � c

i

, where Gx

t1(y) := �⇡(0, y) (1� F (xt

1

| y)) + �⇡(1, y)F (xt

1

| y) is depictedin Figure 4.

When y is small, this function is equal to�⇡(1, y), then jumps continuously from�⇡(1, y)to �⇡(0, y) when y is in the � neighborhood of xt

1

and then it becomes equal to �⇡(0, y).

11

Figure 4: The function Gx

t1(y). The vertical line is xt

1

.

We may consider then the equation:

c = Gx

t1(y) .

Thus, given xt

1

the solution xt

i

of (7) and (8) (resp.) is either on the jump or beyond it (ifit was before we would get as solutions k

i

which can not be). If the solution is beyond thejump, we get a similar conclusion as above; we have gone all the way to k

i

and thus thesequence would have collided with xt

i

and we would have already isolated the equilibriumstrategy for player i, so we assume that on each t, the solution is in the jump and denotethis solution as Y (xt

1

, c). Then

Y�xt

1

, c�=

dxt

1

+ 4c� � 3d�

d+ 4m�

and so

xt

2

= Y�xt

1

, c2

�

=dxt

1

+ 4c2

� � 3d�

d+ 4m�(9)

xt

3

= Y�xt

1

, c3

�

=dxt

1

+ 4c3

� � 3d�

d+ 4m�. (10)

Now, plugging (9) and (10) into (6) we get xt

1

as a function of xt�1

1

.

xt

1

=d⇣

dx

t�11 +4c2��3d�

d+4m�

+ dx

t�11 +4c3��3d�

d+4m�

+ 2�⌘� 4c�

2 (d� 2m�)

=d2xt�1

1

+ 2 (c3

� c1

+ c2

� d) d� + 4 (d� 2c1

)m�2

(d� 2m�) (d+ 4m�).

12

Taking the limit when t ! 1 we get

x11

=d2x1

1

+ 2 (c3

� c1

+ c2

� d) d� + 4 (d� 2c1

)m�2

(d� 2m�) (d+ 4m�).

which gives

x11

=(c

3

� c1

+ c2

� d) d+ 2 (d� 2c1

)m�

m (d� 4m�)

We can now calculate x12

and x13

:

x12

= Y (x11

, c2

)

x13

= Y (x11

, c3

)

If we take � ! 0 we obtain that for i 2 {1, 2, 3}, x1i

! c3�c1+c2�d

m

and if c3

� c1

> d

2

, we getthat

lim�!0

x12

=c3

� c1

+ c2

� d

m>

d

2

+ c2

� d

m=

2c2

� d

2m= k

2

.

So given (c1

, c2

, c3

, d,m), there exists a threshold �b

, that depends on (c1

, c2

, c3

, d,m), suchthat if � is smaller than �

b

, then the sequence starting from below for player 2 converges toa point strictly greater than k

2

.By the analogous exercise developed from above, we will get that given (c

1

, c2

, c3

, d,m),there exists a threshold �

a

, that depends on (c1

, c2

, c3

, d,m), such that if � is smaller than�a

, then the sequence starting from above for player 2 converges to a point strictly smallerthan k

2

.So if � is smaller than min {�

a

, �b

}, then the only strategy of player 2 isolated by theprocess of iterated elimination is her unique equilibrium strategy, implying that for all threeplayers the only strategies isolated by the process of IESDS are their equilibrium strategies.

13