Lecture 24: Game Theoryc 2009 Je¤rey A. MironOutline1. Introduction2. The Payo¤ Matrix of a Game3. Nash Equilibrium4. Mixed Strategies5. The Prisoner’s Dilemma6. Repeated Games7. Sequential Games8. A Game of Entry Deterrence1 IntroductionIn the previous lecture we discussed oligopoly behavior, in particular, the fact thatcartel members have an incentive to cheat. This creates a demand for strategies thecartel members could pursue that might discourage cheating and increase pro…ts onaverage.This is but one example of a more general phenomenon: strategic interactionoccurs in many aspects of life. By strategic interaction, we mean situations in whichmy desired choice of behavior depends on what I think you will do, and your desiredchoice depends on what you think I will do. This kind of situation is addressed bythe branch of economics known as game theory.Note the key di¤erence between the strategic setting and most of those we haveconsidered so far. Both a pure competitor and a monopolist take market conditionsas given; they can then decide what actions are in their own best interest without1worrying about how some other economic actor will respond. In all the models ofconsumer behavior that we have considered, consumers took prices as given.In many instances, however, what matters is how I think others will respond tomy actions, how they think I will respond to their actions, how I think they willthink that I will respond, and so on.We now examine game theory as a method of modeling these kinds of environ-ments. We are interested in applications of game theory to the theory of the …rm,especially in the case of oligopoly, but also in the insights and techniques that comefrom game theory more generally.2 The Payo¤ Matrix of a GameGame theory in principle can b e applied in settings with many participants (e.g., anindustry with many competitors), but we limit our analysis to two-person games forsimplicity. This allows us to p ortray the key aspects of the game in a payo¤ matrix.Consider a speci…c example with two players named A and B. Each player canchoose between two strategies.Player A writes one of two words on a piece of paper, “Top”or “Bottom.”Simultaneously, Player B writes “Left”or “Right.”Each player does this independently, i.e., without knowledge of the other’s actions.After the players do this, the pieces of paper are examined, and each player getsthe payo¤ depicted in the table.2Player APlayer BTopBottomLeft Right1, 2 0, 12, 1 1, 0The payo¤ to A is the …rst entry in each pair; the payo¤ to B is the second entry.Person A has two strategies: top or bottom. These could represent economicchoices like raise price or lower price, or they could represent other things like declarewar or make peace; go to the movies or not, etc. The strategies do not have to bestrategic per se; they are just choices the players might make.Note that this approach forces the set of strategies to be discrete –raise price orlower price –when in reality, the choices available to real economic actors might becontinuous –choose an amount by which to raise or lower price. Whether this isa serious restriction depends on the setting. In some cases, it is reasonable (e.g., inactual games that have sp eci…c rules about the p ossible actions) while in others, itpotentially assumes away economically relevant interactions (e.g., the adjustment ofprice to achieve equilibrium).What will b e the outcome of this game?In this case, a simple solution exists. For A, it always better to say Bottombecause the payo¤ is higher whether B chooses Left or Right. For B, it is alwaysbetter to choose Left because the payo¤ is higher whether A chooses Top or Bottom.Thus equilibrium strategy is for A to play Bottom and for B to play Left.3Note that each player in this game has what is referred to as a dominant strat-egy: a strategy that is better no matter what the other player do es.Whenever one player has a dominant strategy, we expect that player to adoptthat strategy.If both players have a dominant strategy, we assume that the equilibrium outcomeof the game will b e the one determined by those two dominant strategies, as in theexample above.3 Nash EquilibriumMany games, however, do not have dominant strategies. Consider the following:4Table: A Nash EquilibriumPlayer APlayer BTopBottomLeft Right2, 1 0, 00, 0 1, 2In this example, A’s best choice depends on what B does, and B’s best choicedepends on what A does, so no dominant strategy exists for either player.Therefore, consider another criterion for predicting what outcome will occur:Assume A makes the choice that is optimal for him assuming B makes the choicethat is optimal for him, and vice versa.That is, we say that a pair of strategies is a Nash equilibrium if A’s choice isoptimal given B’s choice, and B’s choice is optimal given A’s choice.Remember that neither player knows what the other will do at the time when astrategy is chosen; the moves are simultaneous.But each player can make an educated guess about what the other will do.So, we can interpret Nash equilibrium as a pair of expectations about the otherplayer’s choice such that, when the other player’s choice is revealed, neither playerwants to change their choice of strategies.In this example, (Top, Left) is a Nash equilibrium.5Note that the Cournot equilibrium we examined under oligopoly is an exampleof a Nash Equilibrium.The Nash equilibrium solution concept is appealing, but it has some problems asa metho d of solving games.First, some games have more than one Nash equilibrium. In the previous exam-ple, (Bottom, Right) is also a NE.Second, some games have no Nash equilibria. See the following example:6Table: A Game with No Nash Equilibria in Pure StrategiesPlayer APlayer BTopBottomLeft Right0, 0 0, -11, 0 -1, 3The possibility of many NE or no NE might seem to suggest that game theoryis not so useful; it is presumably awkward to use a theory that either generates noprediction about the equilibrium, or makes multiple, di¤erent predictions. In fact,things are not quite so bleak.4 Mixed StrategiesThe second problem with Nash Equilibrium –the possibility that a game has no NE–can be reduced by broadening our de…nition of strategies.So far we have b een thinking of choices about strategies as discrete choices: playTop or Bottom, Left or Right. These approaches are called pure strategies.Another possibility, however, is that each person