Economic Theory

Axiomatic Models of Bargaining by Alvin E. Roth (auth.)

By Alvin E. Roth (auth.)

The challenge to be thought of this is the only confronted through bargainers who needs to achieve a consensus--i.e., a unanimous selection. in particular, we'll be consid­ ering n-person video games within which there's a set of possible choices, anyone of that are the end result of bargaining whether it is agreed to by way of all of the bargainers. within the occasion that no unanimous contract is reached, a few pre-specified disagree­ ment final result stands out as the consequence. hence, in video games of this kind, each one participant has a veto over any replacement except the war of words final result. There are numerous purposes for learning video games of this sort. First, many negotiating events, really these regarding simply bargainers (i.e., while n = 2), are carried out less than primarily those ideas. additionally, bargaining video games of this kind frequently happen as elements of extra complicated strategies. In addi­ tion, the simplicity of bargaining video games makes them a good motor vehicle for learning the impression of any assumptions that are made of their research. The impact of a few of the assumptions that are made within the research of extra complicated cooperative video games can extra simply be discerned in learning bargaining video games. some of the types of bargaining thought of right here may be studied axioma- cally. that's, each one version can be studied by way of specifying a collection of homes which serve to symbolize it uniquely.

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W . ) l l* i i i l i i h-K) u ( w + C ) . n The Nash solution can now be characterized as selecting the point z* in S* at which the players are equally bold (or equally fearful of ruin). That is, we have the following result. Theorem 7: F(S*,d*) = z* = Ü ^ Ö ^ + c*), u ( w 2 2 + c*)) such that c* + c* = Q and t> (w ,c*) - b (w ,c*). 1 1 2 Proof: 2 The Nash solution picks the point which maximizes the product ACc ) - [u (w + c ) - u (w )][u (w + Q - c ) - u ( w ) ] . ^] + 2 U l/ l W + c l^ 2^ 2 u w + 1 ^ ~ l) C 1 " 2 ^ 2 ^ ~ °* u w w n i c gives us the re­ n quired result.

Z. (c*)). 2 * l i i i i r 1 0 v Then we can state the follow- ing corollary of Theorem 5. 1; If v^ is strictly more risk averse than v^ on the interval 0 < c < Q, then c* < c*. That is, the Nash solution predicts that the player who is more risk averse on the range of feasible monetary payoffs will receive the smaller payoff. Proof: Suppose v^(c) - k(v^(c)) for 0 £ c <_ Q, where k is an increasing con­ cave function. If k were linear, then v^ and v^ would be equivalent utility func­ tions, and the symmetry of Nash's solution would imply c* = c* = Q/2.

If k were linear, then v^ and v^ would be equivalent utility func­ tions, and the symmetry of Nash's solution would imply c* = c* = Q/2. However k is strictly concave, so v^ is strictly more risk averse than v ^ and the corollary now follows immediately from Theorem 5. The Harsanyi-Zeuthen model of negotiation considered in the previous section can be used to provide an alternative, indirect proof of Theorem 5, by means of the following observation. 1: If and are two utility functions such that is more risk averse than u^, then the risk limit r^ for any pair of proposals is lower under v.

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