Info Czj
Exercise 14.3 Consider a game of incomplete information see chapter 6 with prior p over types 0. Suppose that p has full support 0 gt 0 for all 0 . Describe the information structure Q, H , and determine the meet. Exercise 14.4 Prove the claim made in footnote 5. Exercise 14.5 Prove that if E is common knowledge at tu under partitions H but is not common knowledge at co under partitions W, , then there is a player j and a state oj such that player fs knowledge is different at a . See the last...
r yfi y
That is, 7, 0 can be reduced by using the deterministic decision x 0 instead of the random decision x 0 . Because random schemes reduce V0 and V,, and raise C , they yield less utility to the principal A vjm. 0 s vl x 0 , 0 - lt r -l- m, n du Turning things around, transforming a deterministic decision x 0 into a random one x 0 with the same mean for each 0 reduces the principal's welfare. We thus conclude that if the agent's incentive-compatibility constraint for the deterministic allocation...
Dynamic Games Of Complete Information
In the examples we examined in part I, such as the stag hunt, the prisoner's dilemma, and the battle of the sexes, the players choose their actions simultaneously. Much of the recent interest in the economic applications of game theory has been in situations with an important dynamic structure, such as entry and entry deterrence in industrial organization and the time-consistency problem in macroeconomics. Game theorists use the concept of a qame in extensive form to model such dynamic...
Spulber D. Contingent Damages
Admati, A. R., and M. Perry. 1987. Strategic delay in bargaining. Review of Economic Studies 54 .145 364. Akerlof, G. 1970. The market for lemons Qualitative uncertainly and the market mechanism. Quarterly Journal of Economics 84 488-500. Ausubel, L., and R. Deneckere. 1987. One is almost enough for monopoly. Rand Journal of Economics 18 255-274. Ausubel. L., and R. Deneckere. 1989a. Reputation in bargaining and durable goods monopoly. Et onometrica 57 511 531. Ausubel, L and R. Deneckere....
Existence of a PureStrategy Equilibrium
With discontinuous payoffs, a compact strategy space no longer ensures that a player's optimal reaction to his opponents' strategies exists. To guarantee existence, we assume that payoff functions are upper semi-continuous. An upper semi-continuous function is a function that has no jumps down. Definition 12.2 A function u on S is upper semi-continuous at s, if, for any sequence sn converging to s, Note that the function u2 depicted in figure 12.4 fails to be upper semi-continuous at p2 pt t 3....
hxistence of a MixedStrategy Equilibrium
ut 7. lt jt gt ut st, ali for all s,- e S and for all i. is satisfied in the Hotelling game. In particular, they require that this sum not jump down in the limit of the equilibrium strategies lim sup w, 7 lt X 12.2 Next, they make the assumption of weakly lower semi-continuous payoffs Let 5 i denote the set of s such that , is discontinuous at s and Assume that discontinuities occur only on a subset of measure 0 in which a player's strategy is related to another player's. That is, for any two...
References Pgy
Banks, J., and J. Sobel. 1987. Equilibrium selection in signalling games. Econometrica 55 647 662. Ben-Porath, E and E. Dekel. 1988. Coordination and the potential for self-sacrifice. Mimeo. Cho. I. K and D. M Kreps. 1987. Signalling games and stable equilibria. Quarterly Journal of Economics 102 179-221. Cho, I. K , and J. Sobel. 1990. Strategic stability and uniqueness in signalling games. Journal of Economic Theory 50 381-413. Dalkey, N. 1953. Equivalence of information patterns and...
A 1
strictly dominated by the truthful report fy Because each player has a dominant strategy, it does not matter whether he knows the other players1 utility parameters. Hence, even if the players do not know one another's payoffs sec chapter 6 , it is still rational for them to tell the truth. This property of the dominant-strategy demand-revelation mechanism called the Groves mechanism makes it particularly interesting in a situation in which a consumer's utility parameter is known only to that...
f ri iti cvr s
is in V. The full-dimension assumption ensures that such t ' i exist for some Again, to avoid the details of public randomizations, assume that for each i there is a pure action profile a i with g a i v' i . Let w ffi mJ denote player f s payoff when minmaxing player j. Choose N such that, for all i max ii Nv lt min g a 5.8 This is the punishment length such that, for discount factors close to 1, deviating once and then being minmaxed for N periods is worse than getting the lowest payoff once...
Info Mgz
Exercise 1.7 public good Consider an economy with I consumers with quasi-linear utility functions, where f, is consumer fs income, x is a public decision for instance, the quantity of a public good , x, gt is consumer i's gross surplus for decision x, and Of is a utility parameter. The monetary cost of decision x is C x . The socially efficient decision is x 0, , fl arg max x, - C x . Assume i that the maximand in this program is strictly concave and ii that for all 0 ,, 0,, and 0 , Condition...
Info Vhk
defines the Nash-cquilibrium correspondence for this one-player game. In particular, for 0, any x e 0,1 is optimal. Figure 1.17, which exhibits the graph of the Nash correspondence in bold , suggests its main , properties. First, the correspondence has a closed graph is upper hemi-continuous . For any sequence ,.x n belonging to the graph of the correspondence and converging to some A, x , the limit ., belongs to the graph of correspondence.20 Second, the correspondence may not be lower...
B
The next example of a correlated equilibrium illustrates the familiar game-theoretic point that a player may gain from limiting his own information if ihe opponents know he has done so, because this may inducc the opponents to play in a desirable fashion. In the game illustrated in figure 2.5, player 1 chooses rows, player 2 chooses columns, and player 3 chooses matrices. In this game the unique Nash equilibrium is D, L, A , with payoffs 1,1,1 . Now imagine that the players build a correlating...
R 1
sider two sequences A - A and an a such that a e r an and a r cr . That is, o is a Nash equilibrium of G A , but a is not a Nash equilibrium of G A . Then there is a player i and a ct, that does strictly better than ot against cr . Since payoffs are continuous in A, for any A near A and any lt r near a , fjj is a strictly better response to than er ' is a contradiction. It is important to note that this does not mean that the correspondence E is continuous. Loosely speaking, a closed graph plus...
q 0
At first, sight a decrease in the number of equilibria might appear to violate the closed-graph property, but this is not the case For positive but small, the mixed-strategy equilibrium A 1 A , X I a is very close to the pure-strategy equilibrium 0,0 . Figures 1.19 and 1.20 display the equilibrium correspondences of these two games. More precisely, for each we display the set of p such that p,q is an equilibrium of N X for some c this allows us to give a two-dimensional diagram. Inspection of...