1Economics 109, Game Theory, Spring 2002, Vincent CrawfordA game is a multi-person decision situation, defined by its players,its "rules" (the order of players' decisions, their feasible decisions ateach point, and the information they have when making them); howplayers' decisions determine the outcome; and players' preferencesover outcomes. I call these things the game's structure.Analyses of games must confront all the issues that arise withindividual decisions, plus one that is unique to games: Because theoutcome is influenced by other players' decisions as well as yourown, to do well in a game you need to predict others' decisions,taking their incentives into account. This may require a mentalmodel of other players (including a model of their models of you).Examples to illustrate issues a theory of games should address:L R Confess Don'tT2212Con-fess-5-5-10-1B2111Don't-1-10-2-2Crusoe vs Crusoe Prisoner's DilemmaCrusoe vs. Crusoe is just two decision problems, not really a game;each player has a best decision independent of other's (dominant).In Prisoner's Dilemma players' decisions affect each other's payoffsbut each still has a dominant decision; individual optimal decisionswith payoff interactions yield a Pareto-inefficient outcome. Don'tneed special theory for games like this (but Prisoner's Dilemmagets more interesting when we study ways to improve its outcome).2PushWaitHeadsTailsPush1553Heads-111-1Wait-1900Tails1-1-11Pigs in a Box Matching PenniesIn Pigs in a Box, Row (R) is a big pig and Column (C) is a little pig.The box is a Skinner box, named for psychologist B.F. Skinner.Pushing a lever at one end yields 10 units of grain at the other.Pushing "costs" either pig 2 units of grain. If R pushes while Cwaits, C can eat 5 units before R lumbers down and shoves Caside. If C pushes while R waits, C cannot shove R aside and Rgets all but one unit of grain. If both push, and then arrive at thegrain together, C gets 3 units and R gets 7. If both wait, both get 0.In experiments with real pigs, if they settle down, it tends to be at (RPush, C Wait). C does better, although R can do anything C cando! This couldn't happen in an individual decision problem. Ithappens here because Wait dominates Push for C, but not for R:the way they interact in this game, only R has an incentive to Push.In games, (the right kind of) weakness can be an advantage! Rmight do better if he could commit himself to giving C more grain ifC Pushed. Understanding this should help to understand manythings in economics. E.g. corporations as legal "persons" have theright to be sued. This is a "right" because it helps enforce contracts.(If the pigs had studied game theory, they wouldn't have to "settledown": they could just figure out that they should play (R Push, CWait). That they got there anyway suggests that learning andrationality arguments yield the same conclusions in the long run.)Matching Pennies has no good pure decisions (often calledstrategies), but a unique good mixed (randomized) strategy. Howwould you play? What if the 1 (–1) for (Heads, Heads) were 2 (–2)?3L M RT075030M052205B705037Dominance-solvable GameThis 3x3 game has a more complex pattern of iterated dominance.L C RT1040331B00102310Domination Via MixedStrategiesThis 2x2 game has dominance only if we consider mixed strategies.L M RT075070M052205B705007Unique Equilibrium withoutDominanceThis 3x3 game has a unique equilibrium combination of strategiessuch that each player's is best for him, given the other's; but nodominance. It shows that we will need a way to analyze players'decisions that takes their interdependence fully into account.4Go Wait Fights BalletGo0011Fights1200Wait1100Ballet0021Alphonse and GastonBattle of the SexesAlphonse and Gaston's problem is that there are two ways to solvetheir coordination problem…and therefore maybe no good way!Each of the two ways requires them to behave differently whenthere are no clues to distinguish their roles.(In the early 1900s Frederick B. Opper created the Alphonse andGaston comic strip, with two excessively polite fellows saying "afteryou, my dear Gaston" or "…Alphonse" and never getting throughthe doorway. They are mostly forgotten, but we have Alphonse-Gaston games in dual-control lighting circuits in our homes.)Alphonse and Gaston Alphonse and Gaston in your home5Coordination games like Alphonse and Gaston show that playersmay have problems even if preferences are the same. If economicsis "about" coordination, we should study such problems—not justthe coordination that happens in competitive markets. (Mixedstrategies can help Alphonse and Gaston learn to coordinate if theyplay the same game over and over; but the mixed strategies servea completely different purpose than in Matching Pennies.)Battle of the Sexes complicates Alphonse and Gaston's problemwith different preferences about how to coordinate (the Hawk-Dovegame from evolutionary game theory is Battle of the Sexes withdifferent labels that highlight the problem of breaking symmetry).How would you play Battle of the Sexes once? Repeatedly? Wouldyou play differently if the 2 for Row at (Fights, Fights) were a 3?Other Player All Other PlayersStagRabbitAll-StagNotAll-StagStag2210Stag20Rabbit0111Rabbit11Two-Person Stag Huntn-Person Stag HuntIn Stag Hunt (Rousseau's story, assembly line, meeting), with twoor n players, there are two symmetric, Pareto-ranked, pure-strategyequilibria, "all-Stag" and "all-Rabbit". There's also an uninterestingmixed-strategy equilibrium. All-Stag is better for all than all-Rabbit;but Stag is riskier in that unless all others play Stag, a player doesbetter with Rabbit. The game is like a choice between participatingin a highly productive but fragile society and autarky, which is lessrewarding but safer because less dependent on coordination.6Terminology and key conceptsThere are two leading frameworks for analyzing games:2 Cooperative game theory assumes rationality, unlimitedcommunication and ability to make agreements. It assumes Pareto-efficiency and sometimes symmetry across players to characterizepossible outcomes of rational bargaining, filtering out many details.2 Noncooperative game theory also assumes rationality, butreplaces the assumptions of unlimited communication and ability tomake agreements with a detailed model of the situation. It usesrationality, augmented by the notion of equilibrium seen above, toexplain outcomes (sometimes including cooperation). Need a cleardistinction