Game Theory Iterated Dominance, Rationalizability

1. Game TheoryIterated Dominance, Rationalizability Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester 2011-12 Week 41 (October 10, 11)

2. Dominated Strategies • A strategy is strictly dominated if there is another (mixed) strategy that is always better, no matter what the opponents are choosing • Formally, strategy sets for player i are denoted by Si and Σi (for mixed strategies); pay-offs by ui(si ,s-i ) • Strategy si Si is dominated if there exists another strategy σ’i Σi s.t. ui(si ,s-i )< ui(σ’i ,s-i ) for all s-i S-i

3. Example I • Why do we explicitly allow for domination by mixed strategies? • M is not dominated by a pure strategy, but it is strictly dominated by a mixed strategy, e.g., by (½,0,½) • Thus, M is strictly dominated

4. Clarification II • Why do we not explicitly allow for other players playing mixed strategies? • If a strategy is dominated for all pure strategies of the other players, then it is also dominated for all mixed strategies • Vice versa, if a strategy is dominated for all mixed strategies of the other players, then it is also dominated for all pure strategies (trivial)

5. Example II – mixed strategies can also be dominated • Clearly, a mixed strategy given positive weight to a dominated pure strategy is itself dominated • But, other mixed strategies can be strictly dominated as well • None of the pure strategies is dominated here, but the mixed strategy (½,½,0) is strictly dominated by D

6. Example III – Second price sealed-bid auction • Suppose there are I bidders, each with valuation vi for the object. • Suppose here valuation is known. • Sealed bid implies it is a simultaneous move game, si =bi; second price means highest bidder wins and pays second-highest bid • Bidding your valuation is dominant strategy, i.e., all other strategies are dominated by si =vi • Proof

7. Iterated dominance • Let S0i = Si and Σ0i = Σi and define iteratively all strategies that are not dominated yet as Sni = {si Sn-1i  there is no σi Σn-1i s.t. ui(σi ,s-i ) > ui(si ,s-i ) for all s-i Sn-1-i } • Si = nSni • Σi is set of all mixed strategies σi such that there does not exists a σ’i withui(σ’i ,s-i ) > ui(σi ,s-i ) for all s-i  S-i • The game is dominance solvable if for all players Si and thus contains only one element • If game is finite, a finite number of iterations will suffice • Justification in terms of common knowledge of rationality

8. (Iterated) Weak dominance (IEWDS) • Same procedure can be applied to weak dominance • Formally, strategy si Si is weakly dominated if there exists another strategy σi Σi s.t. ui(si ,s-i ) ≤ui(σi ,s-i ) for all s-i S-i with strict inequality for some s-i

9. (Iterated) Weak dominance (IEWDS) • Order of elimination may matter for final outcome • E.g., one may eliminate in this order U (weakly dominated by D), then L. Outcome M and D for player 1 and R for player 2 • Or, eliminate in this order M, R and then you have U and D for 1 and L for player 2. • Or, U and M simultaneously. Outcome (D,L) and (D,R)

10. Justification for IEDS clear • Order of elimination for iterated dominance does not matter • Main idea: if some strategy is strictly dominated for a larger set of strategies of the opponent, then it is also dominated for a smaller set. Hence, if one does not immediately delete a strategy, it should be deleted at some point. • Also, one does not need to delete immediately mixed strategies

11. Application to Cournot oligopoly • N firms • Linear market demand P(Q) = a-bQ • Linear cost C(q) = cq • Firms want to max {(P(qi+Q-i) qi – C(qi)}

12. q2 r1 qM q1 Best response functions for 2 firms • What can be eliminated in first round? • everything that is a best response cannot be eliminated. • only possibility is quantities above monopoly output • fact that they are not best reponses does not necessarily imply they are dominated qM

13. Elimination in rounds I • Round 1: All outputs above monopoly level are dominated • π/q = a – b(n-1)q-i – c – 2bqi < 0 for all q-i and qi > (a – c)/2b • thus, monopoly output gives always higher profit than larger output levels • Round 2: for n=2, all output levels smaller than (a – c)/4b (= optimal response to monopoly output) are dominated • π/q = a – bq-i – c – 2bqi > 0 for all q-i < (a – c)/2b and qi < (a – c)/4b

14. Elimination in rounds II • Round 2: for n>2, no output levels are dominated • π/q = a – b(n-1)q-i – c – 2bqi < 0 for q-i = (a – c)/2b and qi > 0 • Hence, there are strategies for the other players that are not dominated such that qi = 0 is a better response than qi > 0. • Round 3 (and more): for n > 2, nothing changes • Round 3 (for n = 2), output levels larger than 3(a – c)/8b are dominated by 3(a – c)/8b • π/q = a – bq-i – c – 2bqi < 0 for q-i > (a – c)/4b and qi > 3(a – c)/8b

15. Elimination in rounds III • Round 4, 6, 8, …: for n=2, again low output levels are dominated • Suppose that [q,q’] are strategies that remain after previous round, with q < qc < q’ < qm and q’ is best response to q • Thus, q’ = (a-c)/2b - q/2 and best response to q’ is (a-c)/2b – q’/2 = (a-c)/4b + q/4 > q if, and only if, q < (a-c)/3b = qc • For any q < qc the following is true π/q = a – bq-i – c – 2bqi > 0 for q-i < q’ = (a-c)/2b - q/2 and qi < (a-c)/4b + q/4 • Hence, strategies qi < (a-c)/4b + q/4 can be eliminated at this round • Similarly for odd rounds for n = 2 • Thus, for n=2 game is dominance solvable

16. Tilman Borgers & Maarten C.W. Janssen • This result is due to the way of replication of the economy, i.e., only the supply side is replicated for fixed demand • Another (more natural?) way to replace is to say that if for a given n, demand is given by P = a – bQ, then we replicate both sides and say that for k = 2,3,4,… there are nk firms and demand is P = a – bkQ • Their main result then says that for general cost and demand functions and for large k, the Cournot game is dominance solvable iff the Cobweb process is dynamically stable

17. Relation between IEDS and naïve learning • Dominance solvability is highly sophisticated “eductive” reasoning process (CKR) • Cobweb process based on very naïve “learning” as also used by Cournot in creating a dynamic story to his theory • Relation in three parts • IEDS and rationalizability • Rationalizability and Cournot dynamics • Cournot for large N and Cobweb

18. Play Guessing Game • Players have to write down a number between 1 and 100 • I calculate average and take 2/3 of it • Player that is closest to 2/3 of the average wins • Which number do you choose? • Analyze the game with dominated strategies

19. Rationalizability • Let Σ0i = Σi and define iteratively for all players iΣni = {σi Σn-1i  there is a σ-i product of the convex hull of Σn-1-i s.t. ui(σi , σ-i ) ui(σ’i, σ-i) for all σ’i  Σn-1i } • The set of rationalizable strategies is Ri = nΣni • Intuitively, a strategy is rationalizable if it is a best response to some belief about the other players’ play, which should be a best response to some belief the other players have about what their opponents will do, etc…. • Convex hull is used to “smoothen the process” and is needed as generally, even though two pure strategies themselves can be a best response, a mixture of them is not • The set of rationalizable strategies is non-empty

20. Rationalizability and IEDS • If a strategy is dominated, it is never a best response and therefore cannot be rationalizable. Thus, rationalizability is a stronger notion • For N=2, the two actually coincide • Graphical illustration • For N > 2, rationalizability is stronger • Exercise 2.7 FT