Tips for Bargaining at Markets_8 docx

Recall that ps1 which depends on δ is the offer made by the seller in the unique subgame perfect equilibrium of the game in which there is asingle buyer; p B is the offer made by the sel

examination the assumption that agents discount future payoffs, when bined with the other assumptions of the model, is not as natural as it seems.The fact that agents discount the future not only makes a delay in reach-ing agreement costly; the key fact in this model is that it makes holding

com-a specicom-al relcom-ationship costly A buyer com-and com-a seller who com-are mcom-atched com-areforced to separate at the end of the bargaining session even if they have aspecial “personal relationship” The chance that they will be reunited isthe same as the chance that each of them will meet another buyer or seller.Thus there is a “tax” on personal relationships, a tax that prevents theformation of such relationships in equilibrium It seems that this tax doesnot capture any realistic feature of the situations we observe

We now try to separate the two different roles that discounting plays inthe model Remove the assumption that pairs have to separate at the end

of a bargaining session; assume instead that each partner may stay withhis current partner for another period or return to the pool of agents wait-ing to be matched in the next period Suppose that the agents make thedecision whether or not to stay with their current partner simultaneously.These assumptions do not penalize personal relationships, and indeed theresults show that noncompetitive prices are consistent with subgame per-fect equilibrium

The model is very similar to that of Section 9.4.2 Here the proposer

is selected randomly, and the seller may switch buyers at the beginning ofeach period In the model of Section9.4.2the agents take turns in makingproposals and the seller may switch buyers only at the beginning of a period

in which her partner is scheduled to make an offer The important feature

of the model here that makes it similar to that of Section9.4.2rather thanthat of Section9.4.1is that the seller is allowed to leave her partner after

he rejects her offer, which, as we saw, allows the seller to make what iseffectively a “take-it-or-leave-it” offer

As in Section 9.4.2 we can construct subgame perfect equilibria thatsupport a wide range of prices Suppose for simplicity that there is a singleseller (and an arbitrary number B of buyers) For every p∗s such thatps(1) ≤ p∗s ≤ ps(B) we can construct a subgame perfect equilibrium inwhich immediate agreement is reached on either the price p∗s, or the price

p∗b satisfying p∗b = δ(p∗s+ p∗b)/2, depending on the selection of the firstproposer In this equilibrium the seller always proposes p∗s, accepts anyprice of p∗

b or more, and stays with her partner unless he rejected a price

of at most p∗

s Each buyer proposes p∗

b, accepts any price of p∗

s or less, andnever abandons the seller

Recall that ps(1) (which depends on δ) is the offer made by the seller

in the unique subgame perfect equilibrium of the game in which there is asingle buyer; p (B) is the offer made by the seller when there are B buyers

and partners are forced to separate at the end of each period The limits

of ps(1) and ps(B) as δ converges to 1 are 1/2 and 1, respectively Thuswhen δ is close to 1 almost all prices between 1/2 and 1 can be supported

as subgame perfect equilibrium prices

Thus when partners are not forced to separate at the end of each period,

a wide range of outcomes—not just the competitive one—can be supported

by market equilibria even if agents discount the future We do not claimthat the model in this section is a good model of a market Moreover,the set of outcomes predicted by the theory includes the competitive one;

we have not ruled out the possibility that another theory will isolate thecompetitive outcome However, we have shown that the fact that agentsare impatient does not automatically rule out noncompetitive outcomeswhen the other elements of the model do not unduly penalize “personalrelationships”

10.5 Market Equilibrium and Competitive Equilibrium

“Anonymity” is sometimes stated as a condition that must be satisfied inorder for an application of a competitive model to be reasonable We haveexplored the meaning of anonymity in a model in which agents meet andbargain over the terms of trade As Proposition8.2shows, when agents areanonymous, the only market equilibrium is competitive When agents havesufficiently detailed information about events that occurred in the past andrecognize their partners, then noncompetitive outcomes can emerge, eventhough the matching process is anonymous (agents are matched randomly).The fact that this result is sensitive to our assumption that there is nodiscounting can be attributed to other elements of the model, which inhibitthe agents’ abilities to form special relationships In our models, matchesare random, and partners are forced to separate at the end of each period

If the latter assumption is modified, then we find that once again specialrelationships can emerge, and noncompetitive outcomes are possible

We do not have a theory to explain how agents form special relationships.But the results in this chapter suggest that there is room for such a theory

in any market where agents are not anonymous

Notes

This chapter is based onRubinstein and Wolinsky (1990)

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