Introduction to Game Theory 3a. More on Normal-Form Games Dana Nau University of Maryland Nau: Game Theory 1 More Solution Concepts Last time, we talked about several solution concepts Pareto optimality Nash equilibrium Maximin and Minimax Dominance Rationalizability We’ll continue with several more Trembling-hand perfect equilibrium ε-Nash equilibrium Rationalizability Evolutionarily stable strategies Nau: Game Theory 2 Trembling-Hand Perfect Equilibrium A solution concept that’s stricter than Nash equilibrium “Trembling hand”: Requires that the equilibrium be robust against slight errors or “trembles” by the agents I.e., small perturbations of their strategies Recall: A fully mixed strategy assigns every action a non-0 probability Let S = (s1, …, sn) be a mixed strategy profile for a game G S is a (trembling hand) perfect equilibrium if there is a sequence of fully mixed-strategy profiles S0, S1, …, that has the following properties: k lim k→∞ S = S k k k k k for each S = (s1 , …, si , …, sn ), every strategy si is a best response to k the strategies S−i The details are complicated, and I won’t discuss them Nau: Game Theory 3 ε-Nash Equilibrium Another solution concept Reflects the idea that agents might not change strategies if the gain would be very small Let ε > 0. A strategy profile S = (s1, . , sn ) is an ε-Nash equilibrium if, for every agent i and for all strategies siʹ ≠ si, ui (si , S−i ) ≥ ui (siʹ, S−i ) − ε ε-Nash equilibria always exist Every Nash equilibrium is surrounded by a region of ε-Nash equilibria for any ε > 0 This concept can be computationally useful Algorithms to identify ε-Nash equilibria need consider only a finite set of mixed-strategy profiles (not the whole continuous space) Because of finite precision, computers generally find only ε-Nash equilibria, where ε is roughly the machine precision Nau: Game Theory 4 Problems with ε-Nash Equilibrium For every Nash equilibrium, there are ε-Nash equilibria that approximate it, but the converse isn’t true There are ε-Nash equilibria that aren’t close to any Nash equilibrium Example: the game at right has just one Nash equilibrium: (D, R) We can use strategy elimination to get it: • D dominates U for agent 1 • On removing U, R dominates L for agent 2 (D, R) is also an ε-Nash equilibrium But there’s another ε-Nash equilibrium: (U, L) In this equilibrium, neither agent’s payoff is within ε of the agent’s payoff in a Nash equilibrium Problem: In the ε-Nash equilibrium (U, L), agent 1 can’t gain more than ε by deviating But if agent 1 deviates, agent 2 can gain more than ε by best-responding to agent 1’s deviation Nau: Game Theory 5 Problems with ε-Nash Equilibrium Some ε-Nash equilibria are very unlikely to arise Agent 1 might not care about a gain of ε/2, but might reason as follows: • Agent 2 may expect agent 1 to to play D since it dominates U • So agent 2 is likely to play R • If agent 2 plays R, agent 1 does much better by playing D rather than U In general, ε-approximation is much messier in games than in optimization problems Nau: Game Theory 6 Rationalizability A strategy is rationalizable if a perfectly rational agent could justifiably play it against perfectly rational opponents The formal definition is complicated Informally, a strategy for agent i is rationalizable if it’s a best response to some beliefs that agent i could have about the strategies that the other agents will take But agent i’s beliefs must take into account i’s knowledge of the rationality of the others. This incorporates • the other agents’ knowledge of i’s rationality, • their knowledge of i’s knowledge of their rationality, • and so on ad infinitum A rationalizable strategy profile is a strategy profile that consists only of rationalizable strategies Nau: Game Theory 7 Heads Tails Example Heads 1,–1 –1, 1 Matching Pennies Agent 1’s pure strategy Heads is rationalizable Tails –1, 1 1,–1 Let’s look at the chain of beliefs For agent 1, Heads is a best response to agent 2’s pure strategy Heads, … … and believing that 2 would also play Heads is consistent with 2’s rationality, for the following reasons 2 could believe that 1 would play Tails, to which 2’s best response is Heads; … … and it would be rational for 2 to believe that 1 would play Tails, for the following reasons: • 2 could believe that 1 believed that 2 would play Tails, to which Tails is a best response; … Nau: Game Theory 8 Strategies that aren’t rationalizable Prisoner’s Dilemma Strategy C isn’t rationalizable for agent 1 3, 3 0, 5 It isn’t a best response to any of agent 2’s strategies 5, 0 1, 1 The 3x3 game we used earlier M is not a rationalizable strategy for agent 1 It is a best response to one of agent 2’s strategies, namely R But there’s no belief that agent 2 could have about agent 1’s strategy for which R would be a best response Nau: Game Theory 9 Comments The formal definition of rationalizability is complicated because of the infinite regress But we can say some intuitive things about rationalizable strategies Nash equilibrium strategies are always rationalizable So the set of rationalizable strategies (and strategy profiles) is always nonempty In two-player games, rationalizable strategies are simply those that survive the iterated elimination of strictly dominated strategies In n-agent games, this isn’t so Rather, rationalizable strategies are those that survive iterative removal of strategies that are never a best response to any strategy profile by the other agents Example: the p-beauty contest Nau: Game Theory 10 The p-Beauty Contest At the start of my first class, I asked you to do the following: Choose a number in the range from 0 to 100 Write it on a piece of paper, along with your name In a few minutes, I’ll ask you to pass your papers to the front of the room After class, I’ll compute the average of all of the numbers The winner(s) will be whoever chose a number that’s closest to 2/3 of the average I’ll announce the results in a subsequent class This game is famous among economists and game theorists It’s called the p-beauty contest I used p = 2/3 Nau: Game Theory 11 The p-Beauty Contest Recall that in n-player games, Rationalizable strategies are those that survive iterative removal of strategies that are never a best response to any strategy profile by the other agents In the p-beauty contest, consider the strategy profile in which everyone else chooses 100 Every number in the interval [0,100) is a best response Thus every number in the interval [0,100) is rationalizable Nau: Game Theory 12 Nash Equilibrium for the p-Beauty Contest Iteratively eliminate dominated strategies All numbers ≤ 100 => 2/3(average) < 67 => any strategy that includes numbers ≥ 67 isn’t a best response to any strategy profile, so eliminate it The remaining strategies only include numbers < 67 => for every rationalizable strategy profile, 2/3(average) < 45 => any strategy that includes numbers ≥ 45 isn’t a best response to any strategy profile, so eliminate it Rationalizable strategies only include numbers < 45 => for every rationalizable strategy profile, 2/3(average) < 30 . The only strategy profile that survives elimination of dominated strategies: Everybody chooses 0 Therefore this is the unique Nash equilibrium Nau: Game Theory 13 p-Beauty Contest Results (2/3)(average) = 21 winner = Giovanni Nau: Game Theory 14 Another Example of p-Beauty Contest Results Average = 32.93 2/3 of the average = 21.95 Winner: anonymous xx Nau: Game Theory 15 We aren’t rational Most of you didn’t play Nash equilibrium strategies We aren’t game-theoretically rational agents Huge literature on behavioral economics going back to about 1979 Many cases where humans (or aggregations of humans) tend to make different decisions than the game-theoretically optimal ones Daniel Kahneman received the 2002 Nobel Prize in Economics for his work on that topic Nau: Game Theory 16 Choosing “Irrational” Strategies Why choose a non-equilibrium strategy? Limitations in reasoning ability • Didn’t calculate the Nash equilibrium correctly • Don’t know how to calculate it • Don’t even know the concept Hidden payoffs • Other things may be more important than winning › Want to be helpful › Want to see what happens › Want to create mischief Agent modeling (next slide) Nau: Game Theory 17 Agent Modeling A Nash equilibrium strategy is best for you if the other agents also use their Nash equilibrium strategies In many cases, the other agents won’t use Nash equilibrium strategies If you can forecast their actions accurately, you may be able to do much better than the Nash equilibrium strategy I’ll say more about this in Session 9 Incomplete-information games Nau: Game Theory 18 Evolutionarily Stable Strategies An evolutionarily stable strategy (ESS) is a mixed strategy that’s “resistant to invasion” by new strategies This concept comes from evolutionary biology Consider how various species’ relative “fitness” causes their proportions of the population to grow or shrink For us, an organism’s fitness = its expected payoff from interacting with a random member of the population An organism’s strategy = anything that might affect its fitness • size, aggressiveness, sensory abilities, intelligence, … Suppose a small population of “invaders” playing a different strategy is added to a population The original strategy is an ESS if it gets a higher payoff against the mixture of the new and old strategies than the invaders do Nau: Game Theory 19 r r' Evolutionary