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Unformatted text preview: GAME THEORY Lecture 12: Review Instructor: Nuno Limão MIDTERM FORMAT 60 minutes Covers all lectures 5 questions: 20 points each Question format similar to Problem sets and class exercises No make‐ups except under the exceptions allowed by university (see syllabus) review_midterm1 1 DEFINING A GAME & ITS CORE ELEMENTS Definition of a game: A game is a formal representation of a situation where individuals (players) interact in a situation of strategic interdependence, i.e. a situation where (i) an individual’s payoff depends not just on his/her own actions but also on those of others and (ii) an individual’s best actions may depend on the actions he expects others to take. Core elements of a situation of strategic interdependence The players, i.e. who is involved The rules, i.e. who moves when, who knows what and when, what are their possible actions The outcomes, i.e. for each possible set of actions, what is the outcome The payoffs, i.e. what are the player’s preferences and thus the resulting utility over each possible outcome review_midterm1 2 PLAYER PREFERENCES, RATIONALITY, BELIEFS AND COMMON KNOWLEDGE Two basic characteristics of players typically necessary to understand behavior and make any predictions: Preferences (represented by a utility function) Players’ beliefs (about what others will do) Common knowledge: a property of player’s beliefs that states they have mutual knowledge about a piece of information (e.g. it rained yesterday) and that all players know that all players have that mutual knowledge, and so on. Rationality: players are rational if they act to maximize their own payoffs given their beliefs about the environment and what other players will do review_midterm1 3 MODELLING STRATEGIC INTERDEPENDENCE: STRATEGIC FORM Definition: A strategy (s) is a fully specified decision rule for how to play a game at any possible situation. Definition: A strategic form representation of a game describes the number of players (I), their strategy set (Si) and the payoff functions (Vi) used to determine the payoff associated with each strategy profile (or outcome). Review examples of modeling situations into strategic form, e.g. app development 2 identical firms simultaneously choosing to develop a set of applications for either Ipad or a tablet with a different operating system Value of application is higher if there are others for same system Firm 2 Firm 1 Ipad apps
Other apps Ipad apps Other apps 1,1
0,0 0,0
1,1 review_midterm1 4 MODELLING STRATEGIC INTERDEPENDENCE: EXTENSIVE FORM Definition: The extensive form of a game is a graphical representation of a situation of strategic interdependence in the form of a “game tree” that depicts (i) the sequence in which players decide, (ii) the actions available to each player when deciding; (iii) the information set of each player whenever a decision is required and (iv) the payoffs each player assigns to each outcome of the game Extensive form representation of pure coordination game w/ simultaneous moves review_midterm1 5 Extensive representation of pure coordination game with sequential moves Firm 1 chooses which to develop (without knowing exactly what 2 will do) Then firm 2 chooses Only difference is that we remove common information set since firm 2 knows which node it is in review_midterm1 6 Better understanding of strategies in more complex games. Recall the definition: A strategy (s) is a fully specified decision rule for how to play a game at any possible situation. Example of strategies in simultaneous coordination game Firm 1 only has one decision node at which to decide action (i.e. one information set) and only two actions possible. So firm 1 has two feasible strategies, each of the possible actions: Ipad, or other A strategy set (collection of all feasible strategies) that is simply S1={Ipad, Other} Firm 2: same S2={Ipad, Other} Example of strategies in sequential coordination game Firm 1 is the first mover and still has only one decision node at which to decide action (i.e. one information set) and so its strategy set is the same as under the simultaneous game: S1={Ipad, Other} Firm 2 now has two distinct situations at which to decide between two actions so it has four feasible strategies… review_midterm1 7 RELATIONSHIP BETWEEN EXTENSIVE AND STRATEGIC FORMS Every extensive form game has a unique strategic form representation A game in strategic form can be represented by one or more extensive forms (non‐uniqueness is because strategic form contains less information) Examples: moving from extensive to strategic in simultaneous coordination game Recall each firm has only two feasible strategies so represented in 2x2 matrix Firm 2 Ipad apps → Firm 1 Other apps 1,1 0,0 0,0 1,1 Ipad apps Other apps review_midterm1 8 Examples: moving from extensive to strategic in sequential coordination game firm 1 has only 2 strategies so two rows Firm 2 has 4 strategies (#actions^# information nodes) so need 4 columns Firm 2 Firm 1 Ipad apps
Other apps Ipad (1)
Ipad (2)
1,1
0,0 Ipad (1) Other (1)
Other (2) Ipad (2) 1,1
0,0
1,1
0,0 Other (1)
Other (2) 0,0
1,1 review_midterm1 9 DOMINANT AND DOMINATED STRATEGIES Approaches to solving strategic form games Dominant strategy equilibrium Iterative deletion of strictly dominated strategies Nash equilibrium (ch. 4) Start with the one requiring least restrictive assumptions (dominant strategy) and if/when we get many predictions, we will require different, more restrictive assumptions. review_midterm1 10 Definition: A strategy s’’ is strictly dominated by s’ if the payoff from s’’ is strictly lower than that from s’ for any strategies played by the other players. Definition: A strategy s’ strictly dominates by s’’ if the payoff from s’ is strictly higher than that from s’’ for any strategies played by the other players. Definition: A strategy s’ is the dominant strategy if it strictly dominates all other strategies s’’ different from s’. Example: cigarette advertising For PM, spending $15m strictly dominates $10 and $5 since payoffs are higher independently of what RJR does. So $15 is a dominant strategy for Philip Morris. Why is it useful to determine if there are dominated and dominant strategies? Rational player never uses a strictly dominated strategy & always uses dominant one review_midterm1 11 Definition: Weak dominance A strategy s’ weakly dominates a strategy s’’ if the payoff from s’ is at least as great as that from s’’ for any strategies chosen by the other players. there are some strategies for the other players whereby the payoff from s’ is strictly greater than that from s’’. Definition: A strategy s’ is weakly dominant if it weakly dominates all other strategies. Suggested exercises: find strategies that are strictly and weakly dominated review_midterm1 12 SOLVING GAMES WHEN PLAYERS ARE RATIONAL Definition Dominant Strategy Equilibrium (DSE): If each player has a dominant strategy, and plays it (as we would expect if they are rational) then the resulting combination of (dominant) strategies and payoffs represent the dominant strategy equilibrium of that game. Are there important games with a DSE? Yes, Prisoner’s dilemma Second price auctions review_midterm1 13 SOLVING GAMES WHEN RATIONALITY IS COMMON KNOWLEDGE Can we predict the outcome/equilibrium of a game if either or both players have no dominant strategy? Yes, e.g. use IDSDS IDSDS procedure Step 1: Delete all strictly dominated strategies from the original game. (assumes each player is rational) Step 2: Delete all strictly dominated strategies from the game that is obtained from the original game after performing step 1. (assumes each player believes they are rational and so is the other) Step 3: Delete all strictly dominated strategies from the game that is obtained from the game derived after performing step 2. (assumes each player believes that all players believe that all are rational) … Step t: Delete all strictly dominated strategies from the game that is obtained from the game derived after performing step t‐1. Stop when no more strategies are strictly dominated. The remaining strategies of this process are said to survive IDSDS Definition: A game is “dominance solvable” if a single strategy survives the IDSDS. review_midterm1 14 Suggested exercises: Find strategies that survive IDSDS … Step 1: For Player 1, “b” is strictly dominated by “d.” review_midterm1 15 Step 1 (ctd): For Player 2, “y” is strictly dominated by “z.” Step 1 (ctd): Reduced game after one round of IDSDS (“b” and “y.” eliminated) Exercise Can we delete any more strictly dominated strategies from subgame above? Is the game dominance solvable? review_midterm1 16 RATIONALIZABLE STRATEGIES Definition: A strategy is rationalizable if it is consistent with rationality being common knowledge Alternative definition: a strategy is rationalizable if it is optimal for a player under at least some beliefs about other players’ strategies that are themselves consistent with rationality being common knowledge. Suggested exercises: Can we rationalize strategy a for player 1? Can we rationalize strategy c for player 1? Player 2 x Player a 3,1
b 1,2
1 c 2,0 d 1,1 review_midterm1 Y z 1,2
0,1
3,1
4,2 1,3
2,0
5,0
3,3 17 NASH EQUILIBRIA: MOTIVATION AND DEFINITION A motivation: If the game is not dominance solvable then we are left with many (possibly all) strategies and thus no prediction about behavior in the game Definition: A strategy profile is a Nash equilibrium if each player’s strategy maximizes their payoff, given the strategies used by the other players Note the two important components required for a Nash equilibrium Players are rational (as before) Beliefs are accurate each player must correctly anticipate the equilibrium strategy used by their opponents Formal definition: Consider a game with n players: 1, 2, ... n. Let Si be player i’s strategy set and let si Si be one of player i’s strategies. Let s*‐i represent strategies for all of the players except i. And so s*‐i has n‐1 elements and s*‐i = (s*1, s*2, …. , s*i‐1, s*i+1, …. s*n). A strategy profile (s*i, s*‐i) is a Nash equilibrium if and only if for all i = 1, 2, … n we have Vi(s*i , s*‐i) Vi(si, s*‐i) for all si Si review_midterm1 18 Definition of a best reply (or best response) A best reply for player i to s‐i is a strategy that maximizes i’s payoff given that the other n‐1 players use strategies s‐i. Formally, s*i is a best response to s‐i if and only if Vi(s*i, s‐i) Vi(si, s‐i) for all si Si. Then we can say that a strategy profile (s*i, s*‐i) is a Nash equilibrium if s*i is a best response to s*‐i for all i = 1, 2, ……, n. review_midterm1 19 NASH EQUILIBRIA: APPLYING BEST RESPONSE TO SOLVE GAMES Considered many classics: Chicken, Stag hunt, coordination, battle of sexes, PD Prisoner’s Dilemma (recall) Are there any Nash equilibria? Yes. Underline the best response for each. Dilbert Ashok Fink
Mum
(Defect) (Cooperate) Fink 10,10 (Defect) Mum ‐25,0 (Cooperate) 0,‐25 ‐3,‐3 What is the relationship between Nash and dominant strategy equilibrium? Same since the dominant strategy is by definition the best response that a player has against anything the other player may do Moreover, if there is a dominant strategy equilibrium there is also a unique NE review_midterm1 20 Using bestreply method in 3player games: matching letters vs. cool new top Each player gets a payoff of 2 if they all wear their lettered T‐shirts (e.g. A,C,E spells a contestant's name) If a player wears non‐lettered t‐shirt but a cool new BEBE top then assured at least 1. Others get zero since they can’t spell anything cool (AC, CA, AE, EA, EC, CE) Exercise: What are the NE? review_midterm1 21 SELECTING AMONG NASH EQUILIBRIA Multiplicity of equilibria undermine predictive power of theory so considerable work devoted to refine or select a subset of such equilibria Selection criterion I (Pareto criterion): Focus on payoff‐dominant Nash equilibrium Find all NE (so no incentive to deviate and consistent w/ individual rationality:) Select the equilibrium for which each player is at least as well off or better Selection criterion II: Focus on undominated Nash equilibria: Find all NE and then focus only on equilibria that do not use weakly dominated strategies Exercises: Find the Nash equilibria of the game below then determine which (if any) are: (i) undominated and (ii) payoff dominant review_midterm1 22 N‐PLAYER DISCRETE GAMES Games with large number of players used to highlight two important effects Tipping: if a large enough number of players adopts one strategy, e.g. standard or convention, than value of others doing the same increases. Congestion: force for playing different strategies Approach: ask if there is a unilateral motive to deviate from alternative outcomes Exercises related to tipping: NE # of symmetric players adopting technology with network effects Exercises related to Congestion: # symmetric players adopting a particular symbol when payoff to having it and being in minority is higher (e.g. Sneetches game) # and identity of firms with asymmetric costs entering industry where profits decrease with # competitors review_midterm1 23 CONTINUOUS GAMES Definition: A continuous game is one where the strategy set is composed of intervals of real numbers, e.g. price is between 0 and 1000. Approach: divide the infinite strategies into subsets and ask for each what are the incentives for each player to deviate. Exercises and applications: Price competition with similar goods Perfect substitutes: duopoly price competition leading to marginal cost pricing Perfect substitutes: Price matching guarantees as a potential “collusion” device Perfect substitutes: spatial market power Imperfect substitutes and market power review_midterm1 24 Using calculus to define best‐reply and finding NE in continuous games Recall: s*i is a best response to s‐i if and only if Vi(s*i, s‐i) Vi(si, s‐i) for all si Si. If the payoff function is concave for any s‐i (so it has a unique maximum, BR(s‐i)) then the best response for i, BRi is defined by the FOC: ∂Vi(BRi, s‐i)/ ∂si = 0 Since a strategy profile (s*i, s*‐i) is a Nash equilibrium if s*i is a best response to s*‐i for all i = 1, 2, …, n, we can find it as the solution to the system of n equations ∂Vi(BRi, s‐i)/ ∂si = 0 for all i. review_midterm1 25 ...
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