Bargaining and Efficiency

Today's lecture was taken by the professor after an interval of a week. The lecture started from the rest of last lecture.


Rest of the last lecture:
This was the case of professional enforcement in a game which there are several shops and customers. Setting r=imidiate punishment cost, δ=discounter, punishment enforcer is easier to sustain in a price equal to marginal cost. It can be say natural monopoly because it is only price competition and if a enforcer enter and down the price, shops go it, and finally, two enforcers share a market in the same price.

These game describes we can reach efficient equilibria by changing our view, not changing situation and resource.


After that, we proceeded new topics:Strategic Bargaining

Strategic Bargaining:
Suppose each player can choose between (i)going directly to the negotiation table and (ii)trying to grab surplus(commit how much he/she want, and if it is possible, he/she can get it). Previously, we assumed efficient and fair negotiation. But this time, we assumed strategic negotiation.


Strategic Grabbing:
Before a game, each player can choose
Suppose each player can choose between (i)going directly to the negotiation table and (ii)trying to grab surplus, and this situation also can be described as game. Sometime, we can see "prisoner's dilemma" or "chiken"(there are two Nash Equilibria, and each choice has risk became lower)

Strategic Commitment

*mathmatical symbols
{A,B}=consists of A and B
[A,B]=consists A to B (contains A and B)
(A,B)=consists A to B (does not contain A or B)
{A}=collection of A


Old result(suppose cost of commitment(c) is zero, probability of commitment success (q) is one):

If commitments are costless and perfect, any feasible payoff can be obtained in equilibrium.(Nash,1953;Ceawford,1982)


Newresult:(suppose c>0, q=1)
With small positive commitment costs and perfect commitment, irerated elimination of strictly dominated strategies leaves only three solutions:
(i)Player 1 gets everything for sure
(ii)Player 2 gets everything for sure
(iii)conflict with large probability

idea of proof: If s<β, w is selected, then other player choose S as 1

New result 2(suppose c>0, q<1)
result 3
With small positive commitment costs and imperfect commitment, iterated elimination of strictly dominated strategies implies conflict with probability q2

Idea of proof:
(a)show that w strictly dominates si<βi
(if si<βi, si-c<βi, therefore each player choose w)

(b)show that mixd strategy which plays 1 with probability si and w with probability 1-si,
now strictly dominates si<(βi,1)
(if bi
(c)1 dominate w
(each players dominate w?)


New result 3:Dynamic bargaining
This is repeated game.
Suppose that failed negotiations continue until agreement.
(i)As long as both negotiators are commited:disagreement
(ii)If both uncommited:Efficient agreement
(iii)If one committed and the other uncommited: the uncommitted negotiator deciders whether to accept proposal or make new commitment

Let c=0, and V denote expected present net value of a player, δ denote discount factor.

V=δ(qqv+q(1-q)v+(1-q)q(1-v)+(1-q)vv)

{qqv,q(1-q)v,(1-q)q(1-v),(1-q)vv}＝{(commit,commit),(commit,uncommit),(uncommit,commit),(uncommit,uncommit)}


V＝δ(1-q2)/2(1－δq2)

if q goes 1, v become 1/2.