A note on linked bargaining

Journal of Mathematical Economics 46 (2010) 238–247

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Journal of Mathematical Economics

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A note on linked bargaining

Zachary CohnStanford University, Department of Mathematics, Stanford, CA 94305, United States

a r t i c l e i n f o

Article history:Received 25 September 2007Received in revised form16 November 2009Accepted 17 November 2009Available online 26 November 2009

PACS:C7

Keywords:Linked bargainingMechanism designBayesian equilibriumEfficiency

a b s t r a c t

A recent result by Jackson and Sonnenschein (2007) describes a general framework for over-coming incentive constraints by linking together independent copies of a Bayesian decisionproblem. A special case of that work shows that if copies of a standard two-player Bayesianbargaining problem are independently linked (players receive valuations and trade simul-taneously on a number of identical copies), then the utility cost associated with incentiveconstraints tends to 0 as the number of linked problems tends to infinity. We improve uponthat result, increasing the rate of convergence from polynomial to exponential and elimi-nating unwanted trades in the limit, by introducing a mechanism that uses a slightly richerand more refined strategy space. Although very much in the same spirit, our declarationsare constrained by a distribution which is skewed away from the expected distribution ofplayer types. When a sufficiently large number of bargaining problems are linked, “truth”is an equilibrium. Moreover, this equilibrium is incentive compatible with the utility costof incentive constraints almost surely equal to 0.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

It is well-known (since Myerson and Satterthwaite (1983)) that for nontrivial Bayesian bargaining problems, there isno incentive compatible mechanism which organizes trades without losing some surplus. A recent paper, Jackson andSonnenschein (2007), discusses a general mechanism for overcoming incentive constraints by linking together a large num-ber of copies of a given decision problem and trading on these problems simultaneously. In this note, we consider the effectsof linking a large number of independent, two-player bargaining problems. By specializing to this setting, we are able toimprove the rate of decay of the loss of surplus and the treatment of participation constraints.

The mechanism presented here has the following desirable properties: truth (suitably defined) is an equilibrium andthe probability that the mechanism fails to optimally arrange trades decays exponentially in the number of games linkedtogether. Consider an infinite sequence of realizations and the linked bargaining problems that result from taking any finiteinitial collection. We call this a nested sequence of realizations. As a consequence of the exponential decay in the inefficiencyof the mechanism, given any nested sequence, there is almost surely a point beyond which all games trade with zero lossof surplus. This compares with the Jackson–Sonnenschein mechanism, which has a polynomial rate of convergence (as theinverse square root of the number of linked games), and almost surely has inefficiencies on some trades, even far out in thesequence.

We consider a mechanism where players make a two-part declaration for each linked game. On the primary declaration,a strict budgeting is enforced: the frequency of each possible declaration is specified by the mechanism, similar to Jacksonand Sonnenschein (2007). However, unlike the treatment there, the constraint does not correspond to the underlying dis-tribution, but is skewed towards a reluctance to trade (i.e., the seller is forced to declare more objects as more valuable

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Z. Cohn / Journal of Mathematical Economics 46 (2010) 238–247 239

than would be expected). By introducing this bias, players can “protect” more valuable items; this prevents the loss asso-ciated with realizations deviating slightly from the expected distribution. However, this increases the likelihood that thisdeclaration does not reflect the actual private valuation of the object. A secondary declaration allows for a correction tothis misrepresentation by indicating whether players are more eager to trade and arranging additional mutually desirabletrades.

Intuitively, it is the common ordering of outcomes for a bargaining game (the price) that allows for a discussion ofreluctance or eagerness to trade and serves as the basis for the mechanism. Both players can be assured of satisfying ex postincentive constraints if they are permitted to declare themselves sufficiently reluctant to trade. The loss associated withthis can be overcome by allowing both players to reveal that they are in fact more eager to trade when appropriate. Thetension between these goals is what is used to determine the extent to which the primary declaration is biased away fromthe expected distribution and is resolved by the secondary declaration.

2. Linked decision problems

We follow closely the setting for linked decision problems given in Jackson and Sonnenschein (2007). For completeness,we restate the basic framework here, specialized to the two-agent bargaining problem.

A two-agent bargaining problem is a quintuple D = (A, S, B, Ds, Db). A denotes a finite set of alternatives, here a tradingprice or the determination that no trade occurs. S and B give possible valuations of the object for the seller and buyerrespectively. For simplicity, each agent’s utility is the value of the object (net of transfers) at the end of the trading period.Ds, Db are probability distributions over the profiles S and B.1

Denote the space of probability distributions on a set X by �(X). A social choice function F : S × B → �(A) for D is amapping from valuations to a probability distribution on outcomes, and represents the target outcome. We say that F or amechanism generally trades efficiently when all and only mutually desirable trades occur at a mutually beneficial price. Wenote here that any such F is ex ante Pareto efficient.2

Given a bargaining problem and a natural number N, a linking mechanism (M, G) with N linked problems is a messagespace M = Ms × Mb together with an outcome function G : M → �(AN). A strategy for a seller is a mapping �N

s : SN → �(Ms)and for a buyer is �N

b: BN → �(Mb).

We adopt the following notation: S = {sj : 1 ≤ j ≤ ms} and B = {bk : 1 ≤ k ≤ mb} where for all j such that 1 ≤ j < ms, wehave 0 < sj < sj+1 and for all k such that 1 ≤ k < mb, we have 0 < bk < bk+1. We also assume that ms, mb ≥ 2.

3. A linked bargaining mechanism

We now define the mechanism for an N-linked Bayesian bargaining game.The declarations of the seller and buyer come from message spaces which are subsets of (S × S)N and (B × B)N , respectively.

Intuitively, the message space for the seller corresponds to a declaration from SN with a possible option on each problem ofindicating being more eager to sell. Similarly, one can think of the buyer making a declaration from BN with a possible optionon each problem of indicating being more eager to buy. We specify these subsets more precisely below, in Eqs. (1)–(4).

We fix the notation f1, f2 and g1, g2 so that the seller’s declaration on the ith copy is (f1(i), f2(i)) and the buyer’s declarationon the ith copy is (g1(i), g2(i)). We call the first and second coordinates of such a pair the primary declaration and the secondarydeclaration, respectively.

The essential idea is to force a budgeting on the primary declaration which is more reluctant to trade than the intrinsicdistribution, and then to add a secondary declaration that can be used to signal an increased willingness to trade. The degreeto which the budget (on primary declarations) is more reluctant is chosen so that truth remains an equilibrium and truepreferences can be revealed (via secondary declarations) with probability close to 1.

Component wise, the mechanism works by trading when either the primary or secondary declarations indicate that atrade is mutually advisable. If a trade occurs, the trading price is the average of the primary declarations if those indicate atrade, and is the average of the secondary declarations otherwise.

More formally, we define the function Mech as follows:

Mech(f1(i), f2(i), g1(i), g2(i)) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

f1(i) + g1(i)2

if f1(i) < g1(i)

f2(i) + g2(i)2

if f1(i) ≥ g1(i) and f2(i) < g2(i)

No Trade otherwise

On the ith copy, this function defines our mechanism’s trading rule, which we write more concisely as Mech(i).3

1 Values are independent, both across a player’s objects and between players. With interdependent values, the main result may not hold.2 Indeed, it is ex post Pareto efficient. The ex ante efficiency is needed so that the game is a decision problem satisfying Theorem 1 of Jackson and

Sonnenschein (2007).3 Defining A so that trades occur at average values is not necessary, but is fixed for concreteness.

240 Z. Cohn / Journal of Mathematical Economics 46 (2010) 238–247

We now describe the restrictions which define the message space. We first give the restriction on the primary declarations:

#{k : f1(k) = sj}N

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Ds(ms) + � + �ms;s if j = ms

Ds(j) + �j;s if 1 < j < ms

Ds(1) − � −ms∑j=2

�j;s if j = 1(1)

Here, � > 0 and �k;s ≥ 0 may depend on N. The �j,s satisfy 0 ≤ �j,s < (1/N) and are used only so that the right hand sideis an integer multiple of (1/N). � denotes the degree to which the declarations are “reluctance-skewed” since it biases theprimary declarations toward the most reluctant type. It is determined in the course of the proofs of Lemmas A.1 and A.3.

We impose similar restrictions on the declarations of the buyer:

#{k : g1(k) = bj}N

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Db(1) + � + �1;b if j = 1

Db(j) + �j;b if 1 < j < mb

Db(mb) − � −mb−1∑

j=1

�j;b if j = mb

(2)

We also restrict the secondary declarations so that they are used only to indicate whether the agent is more eager totrade than the primary declaration indicates or not.

If f1 (i) = sk then f2(i) ∈ {sk−1, sk}.(If f1(i) = s1 then f2(i) = s1.)

Declaring (sk, sk−1) is permitted only if for some j, sk−1 < bj ≤ sk.

(3)

The first condition corresponds to the intuitive notion stated above that the secondary declaration can indicate whetherthe seller is more eager to trade. The third condition is a technical one which is imposed so that the secondary declarationadds information only when it has the potential to affect whether or not trades take place, and not simply trading price.

We impose symmetric conditions on the buyer:

If g1(i) = bk then g2(i) ∈ {bk, bk+1}.(If g1(i) = bmb

then g2(i) = bmb.)

Declaring (bk, bk+1) is permitted only if for somej, bk ≤ sj < bk+1.

(4)

Note 1. We recover a mechanism which approximates the one in Jackson and Sonnenschein (2007) by ignoring thesecondary declaration altogether and setting � = 0, so that the restrictions on the message space reflect the underlyingdistribution.

A declaration (f1(i), f2(i)) ∈ (S × S)N is said to be truthful up to the constraint or simply truthful if the primary declarationsreflect the ordering of private values and the secondary declarations indicate whether the primary declaration is more reluc-tant to trade than the actual private value. Formally, consider a realized private valuation v ∈ SN . Then a truthful declarationsatisfies the following:

• v(i) > v(j) ⇒ f1(i) ≥ f1(j)• f2(i) < f1(i) ⇒ v(i) < f1(i)• v(i) < f1(i) ⇒ f2(i) < f1(i) whenever the seller has the option to choose f2(i) /= f1(i)

These conditions reflect an assignment of the most reluctant primary declarations to the objects the seller values highest,and the use of the secondary declarations to indicate a greater eagerness to trade than the primary declaration reflects whenappropriate. A symmetric definition exists for the buyer. The strategy of randomized truth consists of making a declarationchosen uniformly at random from the set of all truthful declarations.

We say that a declaration by the seller is effective if for all k such that 1 ≤ k ≤ mb, f2(i) < bk if and only if v(i) < bk. Asymmetric definition exists for the buyer. Since f2(i) < g2(i) is a necessary and sufficient condition for a trade to occur onthe ith copy, if both declarations are effective then the mechanism splits the surplus: all and only mutually desirable tradestake place.

Note 2. The restrictions from Eqs. (1)–(4) constrain the secondary declarations. We show in Lemma A.1 that with proba-bility tending to 1 in the number of games linked, these restrictions do not prevent players from being able to make effectivedeclarations.

4. A theorem on linked bargaining

Recall that a mechanism trades efficiently if all and only mutually desirable trades occur at a price between the privatevalues of the seller and buyer.


Theorem 4.1. Fix an infinite, nested sequence of realizations corresponding to N-linked games for every N. There exists � > 0such that for all 0 < � < � there exists N such that for all N > N the above defined mechanism on the N-linked bargaining gamehas randomized truth as an equilibrium. For that equilibrium, the fraction of realizations which fail to trade efficiently tends to 0exponentially as N tends to infinity.

The parameter � here is the same one used in defining the mechanism in Eqs. (1) and (2).The exponential nature of the convergence means that we have the following corollary.

Corollary 4.2. Fix an infinite, nested sequence of realizations corresponding to N-linked games for every N. With probability 1,there exists � > 0 such that for all 0 < � < � there exists N such that for all N > N the above defined mechanism on the N-linkedbargaining game has randomized truth as an equilibrium which trades efficiently.

The proofs of Theorem 4.1 and Corollary 4.2 are contained in Appendix A.

5. Refinements and participation constraints

We have assumed above that the parameter � used to define the budgeting restriction is independent of N to highlight thefact that the skew of the budgeting restriction is not an artifact of a small number of linked games. However, the proofs allowfor an � which depends on N, as long as it does not decay too quickly. For example, one may choose �(N) = min((�/2), N−˛) forany 0 < ˛ < (1/2).4 In this case, the proportion of problems on which the primary declaration fails to reflect the underlyingvaluation tends to 0. This allows the mechanism to be more directly compared to that of Jackson and Sonnenschein (2007).

Another refinement guarantees that certain participation constraints are satisfied. Essentially, the mechanism is aug-mented by a second stage where agents have the ability to opt out of participating on all problems after seeing what theresult of the linking mechanism would be, in order to satisfy an ex post participation constraint.5

Acknowledgements

The author would like to thank Hugo Sonnenschein for advice and encouragement and two anonymous referees for theirhelpful comments.

Appendix A.

We now prove the main results. We first show in Lemma A.1 that an effective declaration is exponentially likely for �sufficiently small and a sufficiently large number of games linked – in this case, players are never forced to trade items atprices they find unfavorable. We then use this result to bound the probabilities on the distribution of declarations under thestrategy of randomized truth in Lemma A.2. We finally use these bounds to show that randomized truth is an equilibriumstrategy in Lemma A.3 for � sufficiently small and a sufficiently large number of games linked – the essential ingredient hereis that when � is sufficiently small (so Eq. (A.7) holds), players use the secondary declaration to indicate that they are moreeager to trade and reveal true preferences.

Lemma A.1. There exists � > 0 such that for all 0 < � < � there exists N such that for all N > N, buyer and seller can makeeffective declarations with probability greater than

1 − 4m exp(

− 2N

m2�2

)

where m = max(ms, mb).

We see in the proof that it is sufficient for � to satisfy

� < min(

2(

Dmin

m− 1

N

),

12 + m

(Dmin + min (ms, mb)

N

))

where Dmin = min(min1≤j≤ms Ds(j), min1≤k≤mbDb(k)).

Proof. We first establish bounds on the probability that the seller can make an effective declaration.Given a valuation v ∈ SN , let h(j) = #{i : v(i) = sj}/N reflect the distribution of private values. Let p(j) = #{i : f1(i) = sj}/N

reflect the restriction on declarations coming from the constraint on primary declarations given in Eq. (1). Then in order forthe seller to be able to make an effective declaration, it is necessary and sufficient that

ms∑j=ms−k+1

p(j) ≤ms∑

j=ms−k

h(j) for all k such that1 ≤ k ≤ ms − 1, and (A.1)

4 The requirement that limN→∞�(N)N(1/2) → ∞ is used in all of the main estimates, particularly in the result of Lemma A.1. A slightly stronger restrictionis assumed for convenience in Eq. (A.8) during the proof of Lemma A.3. Any member of the family given here satisfies that restriction as well.

5 The procedure is detailed in the proof of Corollary 1 of Jackson and Sonnenschein (2007).


ms∑j=ms−k

h(j) ≤ms∑

j=ms−k

p(j) for all k such that 0 ≤ k ≤ ms − 1. (A.2)

Let ı = max1≤j≤ms |h(j) − Ds(j)|. We now show that the conditions from A.1 and A.2 are satisfied when ı is sufficientlysmall, and finally demonstrate that ı is sufficiently small with the desired likelihood.

For the first inequality, Eq. (A.1), consider:

ms∑j=ms−k+1

p(j) = � +ms∑

j=ms−k+1

(Ds(j) + �j;s)

≤ � +ms∑

j=ms−k+1

(Ds(j) − h(j) + �j;s) − h(ms − k) +ms∑

j=ms−k

h(j).

As |h(ms − k) − Ds(ms − k)| ≤ ı, it follows that −h(ms − k) ≤ ı − Ds(ms − k). Since 0 < �j;s ≤ (1/N),∑ms

j=ms−k+1�j;s < (k/N).Using these substitutions,

ms∑j=ms−k+1

p(j) ≤ � + (k + 1)ı + k

N− Ds(ms − k) +

ms∑j=ms−k

h(j).

It therefore suffices that � + (k + 1)ı + (k/N) − Ds(ms − k) be nonpositive for all 1 ≤ k ≤ ms − 1:

ı ≤ min1≤k≤ms−1

(Ds(ms − k) − � − (k/N)

k + 1

).

Writing Dmin;s = min1≤j≤ms Ds(j), we have a simpler, sufficient condition

ı ≤ Dmin;s

ms− �

2− 1

N.

For the second inequality, Eq. (A.2), consider:

ms∑j=ms−k

h(j) ≤ms∑

j=ms−k

(Ds(j) + ı)

= −� +ms∑

j=ms−k

(p(j) − �j;s + ı)

= −� + (k + 1)ı − k + 1N

+ms∑

j=ms−k

p(j).

We record this result:ms∑

j=ms−k

p(j) − h(j) ≥ � − (k + 1)ı + k + 1N

. (A.3)

It therefore suffices that −� + (k + 1)ı − ((k + 1)/N) be nonpositive for all 0 ≤ k ≤ ms − 1, giving the sufficient condition

ı ≤ �ms

+ 1N

.

Collecting the above, we see that a seller can make an effective declaration if

ı ≤ min(

Dmin;s

ms− �

2− 1

N,

�ms

).

If we assume in addition that � ≤ (1/(2 + ms))(Dmin;s + (ms/N)), this is simply ı ≤ (�/ms).We now bound the likelihood that ı ≤ (�/ms).The exponential version of Chebyshev’s inequality (see, for example, Shiryaev, 1996, p. 69) gives a quantitative estimate

for Bernoulli’s law of large numbers. Applying that result here yields

Pr(|h(j) − Ds(j)| > ˇ

)< 2e−2Nˇ2

.


Therefore, when ˇ = �/ms,

Pr(

|h(j) − Ds(j)| >�

ms

)< 2 exp

(− 2N

m2s

�2

)

Pr(

max1≤j≤ms |h(j) − Ds(j)| >�

ms

)< 2ms exp

(− 2N

m2s

�2

).

(A.4)

Thus, Pr(

ı ≤ (�/ms))

≥ 1 − 2ms exp(−(2N/m2s )�2).

One can weaken Eq. (A.4) to the inequality

Pr(

max1≤j≤ms |h(j) − Ds(j)| >�

ms

)< 2m exp

(− 2N

m2�2

).

A similar argument shows that the buyer fails to make an effective declaration with probability bounded by2m exp

(−(2N/m2)�2

). Therefore, the probability that either the buyer or the seller is unable to make effectively truthful

declarations is bounded by 4m exp(−(2N/m2)�2

). �

Lemma A.2. There exists � > 0 such that for all 0 < � < � there exists N such that for all N > N, the probability of a seller usingthe strategy of randomized truth making a declaration of (sk, sk) or (sk, sk−1) on a given copy is subject to the following bounds:

Pr(sk, sk) ≥(

1 − 2ms exp

(− 2N

m2s

�2

))(Ds(k) − � − ms − 1

N

)

Pr(sk, sk−1) ≤

⎧⎪⎨⎪⎩

� + ms

N+ 2ms exp

(− 2N

m2s

�2

)Ds(k) if k /= ms

�(

1 + 1ms

)+ 1

Nif k = ms

Similar estimates hold for the distribution of declarations by a buyer using the strategy of randomized truth:

Pr(bk, bk) ≥(

1 − 2mb exp

(− 2N

m2b

�2

))(Db(k) − � − mb − 1

N

)

Pr(bk, bk+1) ≤

⎧⎪⎨⎪⎩

� + mb

N+ 2mb exp

(− 2N

m2b

�2

)Db(k) if k /= 1

�(

1 + 1mb

)+ 1

Nif k = 1

Proof. We adopt the notation and assumptions of Lemma A.1. First, we observe that by definition,

#{i : (f1(i), f2(i)) = (sk, sk)}N

+ #{i : (f1(i), f2(i)) = (sk, sk−1)}N

= p(k)

and so that if a secondary declaration is not permitted for some sk, then we must have Pr(sk, sk−1) = 0 and Pr(sk, sk) = p(k),where p(k) is the restriction on primary declarations given in Eq. (1).

Eqs. (A.1) and (A.2) give the rise to the inequalities

#{i : (f1(i), f2(i)) = (sk, sk)}N

≥ h(k) −ms∑

j=k+1

p(j) − h(j) (A.5)

#{i : (f1(i), f2(i)) = (sk, sk−1)}N

≤ms∑j=k

p(j) − h(j) (A.6)

which hold with equality when (sk, sk−1) is permissible.Now consider a seller playing according to the strategy of randomized truth. We calculate the probability �k;s that this

agent declares (sk, sk) on any given copy of the game. First, we consider the case 1 < k < ms.If the seller is making effective declarations, Eq. (A.5) gives

�k;s ≥ h(k) −ms∑

j=k+1

p(j) − h(j).

Using the bound from Eq. (A.3), it follows that

�k;s ≥ h(k) −(

� − (ms − k + 1)ı + ms − k + 1N

).


By definition, ı = max1≤j≤ms |h(j) − Ds(j)|, and so h(k) ≥ Ds(k) − ı. Using that bound,

�k;s ≥ Ds(k) − ı −(

� − (ms − k + 1)ı + ms − k + 1N

)

and so simplifying that expression, we have

�k;s ≥ Ds(k) − � + (ms − k)ı − ms − k + 1N

�k;s ≥ Ds(k) − � − ms − 1N

as 1 < k < ms.Eq. (A.4) bounds the probability that the seller can make an effective declaration from below by 1 −

2ms exp(−(2N/m2

s )�2)

, and so for 1 < k < ms,

�k;s ≥(

1 − 2ms exp

(− 2N

m2s

�2

))(Ds(k) − � − ms − 1

N

).

In the remaining cases (k = 1 and k = ms), if the seller is making effective declarations then

�1;s = p(1) = Ds(1) − � −ms∑j=2

�j;s

≥ Ds(1) − � − ms − 1N

�ms;s ≥ h(ms) ≥ Ds(ms) − ı

≥ Ds(ms) − �ms

.

Generally,

�1;s ≥(

1 − 2ms exp

(− 2N

m2s

�2

))(Ds(1) − � − ms − 1

N

)

�ms;s ≥(

1 − 2ms exp

(− 2N

m2s

�2

))(Ds(ms) − �

ms

)

and so the bound derived above holds uniformly for all k:

�k;s ≥(

1 − 2ms exp

(− 2N

m2s

�2

))(Ds(k) − � − ms − 1

N

).

Finally, we note that this gives an upper bound on the probability of declarations of the type (sk, sk−1) with respect to thestrategy of randomized truth, which we denote by �k;s. Either �k;s = 0 or it satisfies �k;s = p(k) − �k;s.

In either case, for 1 < k < ms,

�k;s ≤ p(k) − �k;s

≤ Ds(k) + 1N

−(

1 − 2ms exp

(− 2N

m2s

�2

))(Ds(k) − � − ms − 1

N

)

≤ 1N

+ � + ms − 1N

+ 2ms exp

(− 2N

m2s

�2

)Ds(k)

≤ � + ms

N+ 2ms exp

(− 2N

m2s

�2

)Ds(k).

For the remaining cases, �1;s = 0. Either �ms;s = 0 or

�ms;s ≤ p(ms) − h(ms)

�ms;s ≤ p(ms) − Ds(ms) + �ms

≤ Ds(ms) + � + 1N

− Ds(ms) + �ms

≤ �(

1 + 1ms

)+ 1

N.

A similar argument proves symmetric estimates for the buyer. These are collected in the statement of the lemma. �


Lemma A.3. There exists � > 0 such that for all 0 < � < � there exists N such that for all N > N, buyer and seller playingrandomized truth is an equilibrium for the N-linked bargaining game.

We see in the proof that it is sufficient for � to satisfy

� <Dmin�min

4�max

where �min = min1≤j≤ms,1≤k≤mb,sj /= bk

∣∣bk − sj

∣∣ and

�max = max(max1<j≤ms sj − sj−1, max1≤k<mbbk+1 − bk).

Proof. We denote by Pr(br, bs) the probability that the buyer’s declaration takes value (br, bs) on some fixed copy of thegame. Since the declaration corresponds to the strategy of randomized truth, Pr(br, bs) is well-defined and independent ofthe particular copy under consideration.

Let

Fsj (sk, sl, br, bs) ={

Mech(sk, sl, br, bs) if sl < bs

sj otherwise

and let Esj (sk, sl) =∑mb

r,s=1Fsj (sk, sl, br, bs)Pr (br, bs).Note that Esj (sk, sl) gives the expected value of a declaration of (sk, sl) when the seller values the object as sj , computed

with respect to the strategy of randomized truth by the buyer.For a seller’s declaration of truth up to the constraint to be in equilibrium with a buyer’s truthful declaration, it suffices

to show that the following two sets of inequalities hold:

• Type I inequalities (the secondary declaration is truthful): For all j, k such that 2 ≤ j ≤ ms, 1 ≤ k ≤ ms, and there exists lsuch that sj−1 < bl ≤ sj , if sk ≥ sj then Esk (sj, sj) ≥ Esk (sj, sj−1).

The verification of these inequalities demonstrates that the secondary declaration used truthfully is an equilibrium replyto an agent using the strategy of randomized truth: when there is a choice of secondary declaration, it differs from theprimary declaration precisely when the seller values the object less than the primary declaration (and so is more eager totrade). We now denote by ♦ the truthful secondary declaration.

• Type II inequalities (the primary declaration is truthful): If sx > sy and sj > si then Esx (sj,♦) + Esy (si,♦) ≥ Esx (si,♦) +Esy (sj,♦).

Showing that these inequalities hold completes the proof of the lemma, since the argument for the buyer is completelysymmetric.

We first show that the Type I inequalities are satisfied. We consider a declaration with f1(i) = sj .Recall that there exists k such that sj−1 < bk ≤ sj as otherwise, only a single secondary declaration is permitted and

so there is nothing to show. We fix the following notation: bk−1 ≤ sj−1 < bk < · · · < bk+r ≤ sj < bk+r+1 where we do notexclude the possibility that r = 0.

The value of Mech.

Mech (sj, sj) (sj, sj−1)

l, z ≥ k + r + 1 : (bl, bz)bl + sj

2bl + sj

2

(bk+r , bk+r+1)bk+r+1 + sj

2bk+r+1 + sj−1

2

(bk+r , bk+r ) N.T.bk+r + sj−1

2

k ≤ l ≤ k + r − 1 : (bl, bz) N.T.bl + sj−1

2

(bk−1, bk) N.T.bk + sj−1

2(bk−1, bk−1) N.T. N.T.

l ≤ k − 2, z ≤ k − 1 : (bl, bz) N.T. N.T.

The table ranges over all possible declarations by the buyer, with entries representing trading values (or N.T. for No Trade).

By considering the entries where the results of mechanism are sensitive to the secondary declaration, we can computeEsx (sj, sj) − Esx (sj, sj−1) in order to show that this is positive if and only if sx ≥ sj .

Suppose first that sx ≥ sj . As the table above shows, declaring (sj, sj) is always weakly best, and sometimes strictlybetter; as all of the buyer’s types listed occur with positive probability, the expectation is positive. Therefore, Esx (sj, sj) −Esx (sj, sj−1) > 0.


If sx < sj , then truth is not always best: against a declaration of (bk+r , bk+r+1) the seller does better to lie. It suffices toshow, however, that

Pr(bk+r , bk+r)

(bk+r + sj−1

2− sx

)+ Pr(bk+r , bk+r+1)

(bk+r+1 + sj−1

2− bk+r+1 + sj

2

)> 0 (A.7)

and hence it is enough to show that

Pr(bk+r , bk+r)(bk+r − sj−1) − Pr(bk+r , bk+r+1)(sj − sj−1) > 0.

We therefore need to produce a lower bound for Pr(bk+r , bk+r) and a sufficiently small upper bound for Pr(bk+r , bk+r+1).We now establish that the estimates given in Lemma A.2 show that this quantity is positive for any � sufficiently small aslong as N is sufficiently large. It is convenient to assume that N is so large that

mb − 1N

+ 2mb exp

(− 2N

m2b

�2

)< �

mb

N+ 2mb exp

(− 2N

m2b

�2

)Db(k + r) < �

1N

<�2

(A.8)

when the results of Lemma A.2, simplify to give the bounds Pr(bk, bk) ≥ Db(k) − 2� and Pr(bk, bk+1) ≤ 2� for all k.In this case, it is sufficient that (Db(k + r) − 2�)(bk+r − sj−1) − 2�(sj − sj−1) > 0 or

� < Db(k + r)

(bk+r − sj−1

2sj + 2bk+r − 4sj−1

)< Db(k + r)

(bk+r − sj−1

4sj − 4sj−1

). (A.9)

The right hand side is positive, and since there are only finitely many types, we can compute its minimum. We repeatthe procedure from the perspective of the buyer, and taking the minimum of those two values, obtain the claimed sufficientupper bound on �.

Now we prove that the Type II inequalities are satisfied: for a seller who makes declarations shown to be preferable bythe Type I inequalities (denoted by ♦), we need to show that Esx (sj,♦) + Esy (si,♦) ≥ Esx (si,♦) + Esy (sj,♦) whenever x > y andj > i.

We use E to denote

Esx (sj,♦) − Esx (si,♦) − (Esy (sj,♦) − Esy (si,♦)) (A.10)

and need to show that E > 0.This depends on the following calculation:

Fsx (sj, sr, bk, bl) − Fsy (sj, sr, bk, bl) ={

0 ifbl ≥ sr

sx − sy ifsr > bl

Fsx (sj, sj, bk, bl) − Fsy (sj, sj−1, bk, bl) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0 ifbk ≥ sj

sj − sj−1

2ifbl ≥ sj > bk

sx − bl + sj−1

2ifsj > bl ≥ sj−1

sx − sy ifsj−1 > bl

Then, before expectations are taken, Eq. (A.10) can be written:

Fsx (sj,♦, bk, bl) − Fsy (sj,♦, bk, bl) − (Fsx (si,♦, bk, bl) − Fsy (si,♦, bk, bl)) . (A.11)

We now prove that Eq. (A.11) is always nonnegative. Lemma A.3 follows by taking expectations.


Recall that sx > sy and sj > si by hypothesis. We use sp, sq, sr , and st to denote the secondary declarations; those valuesdepend on si, sj , sx, sy, and the message space.

Fsx (sj, sp, bk, bl) − Fsy (sj, sq, bk, bl) − (Fsx (si, sr, bk, bl) − Fsy (si, st, bk, bl)) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 if bk ≥ sj

sp − sq

2if bl ≥ sp > bk

sx − bl + sq

2if sp > bl ≥ sq

sx − sy if sq > bl and bk ≥ si

sx − sy − sr − st

2if sq > bl ≥ sr and si > bk

bl + st

2− sy if sr > bl ≥ st

0 if st > bl

Here, some cases are empty, depending on the secondary declarations; for example, there may not be any b ∈ B satisfyingsp > b ≥ sq.

Since sx > sy, the Type I inequalities imply that sp ≥ sq and sr ≥ st . In addition, if sp > sq then sx ≥ sj > sy (and so sq = sj−1 ≥sy). If sr > st then sx ≥ si > sy (and so st = si−1 ≥ sy). One can then verify that in each case, the expression is nonnegative.

Therefore, if the Type I truth conditions are satisfied, then the strategy of following the Type II truth conditions is weaklydominant. It follows that the strategy pair of randomized truth is an equilibrium. �

Proof of Theorem 4.1. Recall that a player with the strategy of randomized truth makes effective declarations wheneverthe constraints permit and that if both players make effective, truthful declarations then the mechanism organizes trades ifand only if they are mutually beneficial.

Let � be as in Lemmas A.1 and A.3. Then for any 0 < � < �, there exists N such that for N > N, buyer and seller playrandomized truth in equilibrium, and with probability greater than 1 − 4m exp

(−(2N/m2)�2

), make effective declarations.

Therefore, for such � and N, the mechanism trades efficiently. �

Proof of Corollary 4.1. Let XN denote the event that buyer or seller are unable to make effective, truthful declarations whenN copies are linked. We have from Lemma A.1 that for N sufficiently large, Pr(XN) < 4m exp

(−(2N/m2)�2

). In particular,∑∞

N=1Pr(XN) < ∞ and so by the lemma of Borel–Cantelli (see, for example, Shiryaev, 1996, p. 255), the probability that eventXN occurs infinitely often is 0. We have already established in Theorem 4.1 that for a sufficiently large N, randomized truthis an equilibrium. Therefore, with probability 1, there is some N such that for all N > N, buyer and seller playing randomizedtruth do so in equilibrium and make effectively truthful declarations. �

Note 3. Remarks following the statements of Lemmas A.1 and A.3 give sufficient conditions on � in terms of N. In thecontext of the statement of Theorem 4.1 and Corollary 4.2, that dependence can be removed and it is sufficient to take � tosatisfy

� < min(

Dmin

2 + m,

Dmin�min

4�max

)

where m = max (ms, mb), �max = max(

max1<j≤ms sj − sj−1, max1≤k<mbbk+1 − bk

),

Dmin = min

(min

1≤j≤ms

Ds(j), min1≤k≤mb

Db(k)

), and �min = min

sj /= bk

∣∣bk − sj

∣∣ .

References

Jackson, M.O., Sonnenschein, H.F., 2007. Overcoming incentive constraints by linking decisions. Econometrica 75 (1), 241–257.Myerson, R.B., Satterthwaite, M.A., 1983. Efficient mechanisms for bilateral trading. Journal of Econometrics Theory 29 (2), 265–281.Shiryaev, A.N., 1996. Probability, 2nd edition. Vol. 95 of Graduate Texts in Mathematics. Springer-Verlag, New York, translated from the first (1980) Russian

edition by R.P. Boas.

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A note on linked bargaining