Dynamic Games of Complete Information Repeated games Best

Dynamic Games of Complete Information.

Repeated games • Best understood class of dynamic games • Past play cannot influence feasible actions or payoff functions • Building block is called ‘stage game’ - i єI is the finite set of players, Ai are finite action spaces - gi: A→R, are payoff functions where A= - players move simultaneously - ht is history before period t, ht=(a 0, a 1, … at-1), and Ht=(A)t is space of all period-t histories at ≡ - A pure strategy for i is seq. of maps - A mixed strategy for i is seq. of maps , where is a probability distribution over Ai

Finite and infinite repeated games • Finite horizon games are solved using backward induction • Payoff functions for the infinite game G(δ) - , where (1 -δ) is normalization factor - for δ→ 1, we use time average criterion • The discount factor δ (<1) represents probability that the game may end at the end of any period • Thus, probability that the t-th stage will be played is δt

A useful result • Theorem: a. Consider a finitely repeated game. If α* is the Nash equil of the stage game, then the strategies “each player i plays α*i in every period” is a SPNE of the full game b. If α* is unique Nash equil of the stage game, then the above strategies constitute the unique SPNE of the full game • Example: The finitely repeated Prisoner’s Dilemma 1 2 Confess Not confess Confess 0, 0 7, -2 Not confess -2, 7 5, 5

An example: Treasury bills auction • • US Treasury Dept periodically sells securities Sold by auction to large financial institutions Auctions held on a regular basis There are two kinds: - single price auctions (one price for all buyers) - multi price auctions (different prices) • For any one kind of security this is repeated game • Which of the two forms should Treasury use?

Treasury bills auction • Simplifying assumptions: 1. two financial institutions 2. quantity of bills, 100, fixed across auctions 3. buyers can offer two prices & two quantities -prices can be high (h) or low (l) -quantity can be 50 or 75 -profit per security with high / low price are πh/πl, with πl > πh.

Treasury bills auction • If both firms offer a high price, then market price is high and total demand is ≥ 100 • If both firms offer a low price, then market price is low • If one wants to buy at h and other at l, then: - in single price auction price is l - in multi price auction one pays h & the other, l - high bidder gets his full qty, rest goes to rival • If price bids are the same, allocation is proportionate to qty demanded

Treasury bills auction • Note: At any price it is always better to ask for a larger quantity • Therefore we can look at the reduced games • Consider two cases: a. Competitive case where 50πh > 25πl. b. Collusive case where 50πh < 25πl.

Treasury bills auction • Competitive case: - in the single price auction h is a dominant strategy, and the unique Nash equilibrium is (h, h) - in the multi price auction both (h, h) and (l, l) can be Nash equilibria • Collusive case: - in the single price auction the Nash equilibria are (h, l) and (l, h). There is also a mixed strategy - in the multi price auction l is a dominant strategy, and the unique Nash equilibrium is (l, l) Treasury prefers the single price auction!!

Infinitely repeated Prisoner’s Dilemma • Consider the grim trigger strategy: a. Start by playing (n, n) and continue playing it as long as no one confesses b. If anyone confesses, play (c, c) from then on • This is a SPNE • If δ>2/7, then cooperation, (n, n) is sustainable! • Why the contrast with prediction from finitely repeated game?

Infinitely repeated Prisoner’s Dilemma • Two important points: 1. Grim punishments may achieve other behaviors 2. Cooperative behavior is achievable with less severe punishments • Example of point 1: -Start with (n, c). Play (n, c) at even numbered periods and (c, n) at odd ones. If there is deviation, play (c, c) from then on. - Show that above is credible • Example of point 2: - A Forgiving trigger strategy says, play (n, n) and if there is deviation play (c, c) for T periods. Revert to (n, n) - Is this credible? - What happens when future is very important, i. e. δ→ 1?

The Folk Theorem for infinitely repeated games • Let player i’s reservation utility or minmax value be: . This is the min value that his rivals can hold him to • Observation: Player i’s payoff is at least in any Nash equilibrium of the stage game, and repeated game, regardless of the discount factor • Let V be set of feasible payoffs, i. e. if v єV, then there exists aєA, such that g(a)=v • The Folk Theorem: For every feasible payoff vector v with vi > for all players i, there exists a <1 such that for all δє( , 1) there is a Nash equilibrium of G(δ) with payoffs v

Nash-threats Folk theorem • Strategy used in proof of Folk thm: Let g(a)=v. Play ai in all periods until there is a deviation. After a deviation by i (say), all players -i play the minmax profile m-ii which gives i a payoff • The above strategies are not subgame perfect Theorem (Friedman 1971) Let α* be a static equilibrium with payoffs e. Then for any vєV with vi > ei , for all players i, there is a such that for all δ> there is a subgame perfect equil of G(δ) with payoffs v. • Friedman’s conclusion is weaker than Folk theorem. Does subgame perfectness restrict set of equil payoffs?

Another Folk theorem Theorem (Aumann and Shapley 1976) If players evaluate sequences of stage game utilities by the time average criterion, then for any vєV with vi > , there is a subgame perfect equilibrium with payoffs v Idea behind proof: a. Use strategy: Play strategy that gives v as long as there are no deviations. If i deviates play minmax profile m-ii which for N periods, where, b. With the time average criterion, minmaxing a deviator is not costly