Unformatted text preview: L ECTURE 5: SOLVING GAMES WHEN PLAYERS ARE RATIONAL AND KNOW IT Econ 414 Instructor: Nuno Limão OUTLINE Dominant and dominated strategies Solving games when players are rational Solving games when rationality is common knowledge (IDSDS) 2/23 DOMINANT AND DOMINATED STRATEGIES Recall a central objective of GT is to model the situations of strategic interdependence AND solve them to better understand and predict behavior. Approaches to solving strategic form games Different ways to solve such a game (i.e. different solution or equilibrium concepts) 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. 3/23 What is a plausible initial set of assumptions we could use to solve a game? Players are rational, i.e. they act to maximize their utility Players follow Sherlock Holmes’s advice and eliminate “impossible” strategies “How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? ‐‐‐SH quoted in Harrington, p. 55 How do we make this idea operational to actually solve a game? We must define what “impossible” means and how we “eliminate” those strategies We must be clear about whether players have knowledge about other’s rationality (mutual knowledge) or even common knowledge about rationality. A definition of “impossible” strategy: one that is “strictly dominated” by another strategy 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’. 4/23 Example: cigarette advertising Strategic form below depicts profits at different levels of advertising for two firms. Payoff = (total expenditure on cigarettes)* market share minus ad costs, where market share = advertising expenditure share we assume a fixed demand for cigarettes (across advertising levels) 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 5/23 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. Note If s’ weakly dominates s’’ we say that s’’ is weakly dominated by s’ While under some strategies of the other player I will be indifferent between s’’ and s’, under others I will be strictly worse off. So if I am uncertain about what the other player will do and I am “cautious” then I will never play a weakly dominated strategy and will always play a weakly dominant one (if it exists) 6/23 Example of weak dominance: Modified cigarette advertising Modify payoff in cigarette ad example in cell (15,15) for PM to be 30 30,35 Now15 still strictly dominates 5 for PM but only weakly dominates 10 since if RJR plays 15 then PM is indifferent between 10 and 15. Since for PM 15 still weakly dominates all strategies we say it is a weakly dominant strategy. 7/23 Exercise: strict and weak dominance Find the strategies that are strictly dominated and those that are weakly dominated for players 1 and 2 8/23 SOLVING GAMES WHEN PLAYERS ARE RATIONAL Weakest assumption: each player knows their respective payoffs and is rational but does not know which strategy the other players will choose or even if they are rational. What games can we solve to obtain a unique solution (i.e. equilibrium) under this assumption? Those that have a dominant strategy for each player. 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, e.g. Prisoner’s dilemma http://www.gametheory.net/media/Dilbert.wmv 9/23 Prisoner’s Dilemma setup 2 players suspected of being co‐conspirators in a crime brought in and placed in different rooms Told that if one remains mum and the other Finks then the Mum one gets full sentence (25 years in jail) and the other walks off. If both Fink then both get convicted but w/ lower sentence since prosecution is easier 2 players know that there is insufficient evidence to convict on the worst crime if both mum but may still get charged and obtain lower conviction (e.g. 3 years in jail) Dilbert
(Defect) (Cooperate) Ashok Fink ‐10,‐10
(Defect) Mum ‐25,0 (Cooperate) 0,‐25 ‐3,‐3 10/23 Prisoner’s Dilemma solution Dominant strategy for Ashok? Fink since If Dilbert finks then payoff to Ashok from fink is ‐10>‐25 (mum) If Dilbert stays mum then payoff to Ashok from fink is 0>‐3 (mum) Dominant strategy for Dilbert is the same (he would know it if paid attention in class) So if each player is rational the prediction is that both Fink 11/23 Prisoner’s Dilemma: famous game, extensively analyzed since the 1940’s. Why? Captures several situations of strategic interdependence with externalities and illustrates why we may end up with a non‐cooperative outcome, e.g. Nuclear arms race Price wars among oligopolists Trade wars among countries Predicts non‐cooperation in a setting where a better option is clear more generally in economics the PD outcome is not Pareto efficient since both would be better off if they had cooperated with each other. Generated a considerable amount of work trying to understand how to move players to more “cooperative” outcomes (as explored in latter lectures) Nuclear arms control treaties Collusion among oligopolists to maintain prices high International trade agreements (e.g. WTO) 12/23 Exercise: Another application of the use of dominant strategies, 2nd price auctions Auctions are a common way to Sell cars, art, antiques, houses, commodities, livestock, tickets, or anything on ebay Allocate broad band licenses to telecommunication companies, large government contracts, emissions permits, government debt, etc Basic structure in most auctions Highest bid wins the object Payoffs are equal to vp: the value you place on the object/license, etc (v) minus the amount you pay, p The amount paid depends on type of auction, e.g. in a sealed bid 1st price auction each individual i submits a single bid, bi, and if yours is the highest you pay that bid value. How can your optimal strategy (i.e. your bid) ever be independent of how others value the object and how others will bid? In a sealed bid second price auction. 13/23 2nd price auction setup (William Vickrey, Nobel 1996)) 2 players: you and her with valuations for a painting of vy =4 and vh =3 If your bid is higher, by > bh , then you win and must pay the second highest bid, bh If your bids match then object is assigned randomly (coin toss) so your expected payoff is ( vy – by)/2, e.g. if both bid 1 then expected payoff is (4‐1)/2 Auctioneer rules that the minimum bid is 1 and the maximum 5 with increments of 1 Questions What are the feasible strategies for each? What are all your possible payoffs if by =1? 14/23 Strategic form representation of 2nd price auction Outcome Consider payoffs from your valuation (4): this strategy is weakly dominant for you Consider payoffs from your valuation (3): this strategy is weakly dominant for her Thus if both are rational (and cautious) (4,3) is a “compelling” prediction Other interesting properties: equilibrium bids should reveal true valuation ex‐post efficiency: allocation to the person with highest valuation for the object in some cases the revenue is equivalent to that in a sealed‐bid 1st price auction 15/23 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, by following Sherlock Holmes’s advice to eliminate the impossible, i.e. the strictly dominated strategies. But it requires stronger assumptions on rationality, e.g. If one has a dominant strategy (e.g. Nerd) and the other does not (e.g. Frat Boy) then the latter must know that the former is rational If neither has a dominant strategy then rationality must be common knowledge 16/23 Example: Team project with single dominant strategy Two students are to be paired up for a term project. Each student can choose either low, moderate, or high level of effort. Grade is increasing in individual effort but effort has different cost for different students There are two types of students: Nerd and Frat Boy. Higher effort not very costly for nerd so his payoffs increase in own effort. Higher effort is costly for Frat boy but offset by higher reward if it leads from “C” to “B” grade , e.g. (moderate, low) to (moderate, high), but not from “B” to an “A”, (mod, mod) to (hi, mod) Strategic form 17/23 Solution for team project game Dominant strategy for Nerd is high effort. There is no dominant strategy for Frat boy, what should he do? If there is common knowledge about the student type (and so the payoffs) AND Frat boy believes Nerd is rational then he will know Nerd will exert high effort no matter what. So Frat boy safely eliminates all the strictly dominated strategies for Nerd (low and moderate) Conditional on high effort by Nerd, Frat boy’s dominant strategy is low effort (6>5,3) 18/23 Solving a game using Iterated Deletion of Strictly Dominated Strategies (IDSDS) Even if neither player has a strategy that dominates all others (i.e. a dominant strategy) they may still have strategies that strictly dominate some other strategies If players are rational and that is common knowledge then they can successively delete any strictly dominated strategies until none are left 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 19/23 Definition: A game is “dominance solvable” if a single strategy survives the IDSDS. Example of employing IDSDS Step 1: For Player 1, “b” is strictly dominated by “d.” 20/23 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? 21/23 Solution Now, for 1 none is dominated but for 2, “x” is dominated by “z.” So eliminate “x.” Now, for 1, “a” is dominated by “d.” So, we eliminate “a.” and are left with The game is much simpler now but not dominance solvable. [next class we will see an alternative way to find an equilibrium in such a case] 22/23 Caution: IDSDS is only applied to STRICTLY dominated strategies. In principle one could apply this procedure to weakly dominated strategies as well. Problem: standard IDSDS leaves us with all possible Nash equilibria (see next class). If we also eliminate weakly dominated strategies then we may end up eliminating some of these equilibria along the way. Moreover, depending on the ORDER of deletion of the strategies it is possible that we end up with different equilibria. 23/23 ...
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- Spring '08
- strictly dominated strategies, Dominant strategy, 23rd Marine Regiment