Q2. What is the central technique used to extract the optimal equilibrium payoff vector?
The central technique employed is the reduction of the repeated game to a static structure from which can be extracted the optimal equilibria in question (and indeed, equilibria supporting any symmetric equilibrium payoff vector).
Q3. What is the first step in defining the game?
The first step in defining the game is to specify the single-period component game G.The Single-Period GameN identical firms simultaneously choose quantities qi, i= l,..., N, of output to produce.
Q4. What is the main point of Porter’s book?
Porter characterizes the choice of q, p, and T (the latter may be infinite) that yields cartel members the greatest discounted profit, subject to the constraint that the resulting regime must be a sequential equilibrium.
Q5. What is the main idea of the analysis?
The analysis has a strong flavor of dynamic programming, and is in the tradition of the stochastic game literature beginning with Shapley [ 121 as well as the more recent papers of Abreu [I f and Radner,yerson, and Maskin [lo].
Q6. What is the probability measure induced on price space?
A probability measure is induced on price space by equilibrium behavior (given some history); the probability measures the sets R and & give the probabilities of remaining in the reward and unishment states, respectively.
Q7. What is the value of the p’-l?
In period t 3 2, the production level is Q( U’- ‘(w)(p’+ ‘)), which is either Q(C) = 4 or Q(a) = 4 (depending on p’-l), since u and y are the only values that U’-‘(w)($F’)= U(U’-2(w)(p’-2))(p(t- 1)) can assume.
Q8. What is the maximum payoff for player i?
Note that for any 4,YESI2Since c(q) > 0 for all q E S,, the maximum single-period payoff for player i is bounded by K, and consequently the maximum supergame payoff for player E’is bounded by 6K/(l--6).
Q9. what is the maximum payoff for a firm i?
For any q>q*=max(q,, K/(~,(l-6)))~ the maximum supergame payoff that firm i faces if it produces the quantity q, for any aggregate production y E S, for the rest of the firms, is bounded bY6” (q.j(q+y)-c(q)+6K/(l -S,} <6. (K-coy-MKl(1 -S)} CO,and since firm i can always choose to produce q = 0 (and therefore get a supergame payoff of 0), firm i will only consider production quantities in z= [0, q*] n S,.
Q10. What is the simplest way to show that B(co( W)) is compact?
Then B(co( W)) = UqEsE~I:(q.en)xAq)l. S’ mce p(q) is compact for each 9 E S, E, is continuous in U, and 3 is finite, B(co( W)) is compact as required.
Q11. What is the way to produce an equilibrium of the game G(v); 6?
if firms believe that v(p(2)) will occur if p(2) is the realization of the price in the second period, it is optimal for them to produce (r,..., r), an equilibrium of the game G(v; 6).
Q12. What is the simplest way to determine the equilibrium of a game?
More generally (if less intuitively), for any bounded set IV of real numbers, let B(W) c R represent the total payoffs that players could receive in pure strategy equilibria of truncated games in which each firstperiod price is followed by some symmetric payoff drawn from l+‘.
Q13. Who would like to acknowledge helpful conversations with Peter Doyle?
The authors would like to acknowledge helpful conversations with Peter Doyle, Abraham Neyman, Rob Porter, and Hans Weinberger.251 0022-0531/S $3.00Copyright Q 1986 by Academic Press, Inc.
Q14. What is the way to construct a new game?
One can imagine constructing a new game by truncating the discounted supergame as follows: after each first-periodlace the SSE successor by the payoffs associated with that successor.
Q15. What is the way to relax the assumption of a sequential equilibrium?
This fact can be used to slightly relax assumption (A3) by requiring only that the set Q defined there be contained in B, an independent of (ql ,..., qN) E 3”‘.