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1 1 Deep Thought BA 592 Lesson I.2 Formulating Games Why can’t the ant and the caterpillar just get along? One eats grass, the other eats Caterpillars … Oh, I see now. --- by Jack Handey.

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2 2 Lesson overview BA 592 Lesson I.2 Formulating Games Chapter 2 How to Think About Strategic Games Lesson I.2 Formulating Games Are Moves Sequential or Simultaneous? Are Players in Pure Conflict? Is the Game Repeated? Do Players have Full or Equal Information? Are the Rules Fixed? Are Agreements Enforceable? Terms and Background Assumptions Review Problems

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3 3 The Six Characteristics of Games Are Moves Sequential or Simultaneous?Are Moves Sequential or Simultaneous? Are Players in Pure Conflict?Are Players in Pure Conflict? Is the Game Repeated?Is the Game Repeated? Do Players have Full or Equal Information?Do Players have Full or Equal Information? Are the Rules Fixed?Are the Rules Fixed? Are Agreements Enforceable?Are Agreements Enforceable? BA 592 Lesson I.2 Formulating Games Are Moves Sequential or Simultaneous?

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4 4 Games where players choose actions simultaneously are simultaneous- move games. n Examples: Prisoners' Dilemma, Sealed-Bid Auctions. n Must anticipate what your opponent will do now, recognizing that your opponent is doing the same. Games where players choose actions in a particular sequence are sequential move games. n Examples: Chess, Bargaining, and Negotiations. n Must look ahead in order to know what action to choose now. n Many sequential move games have deadlines or time limits on moves. n It may be better to move first (chess, …) or second (poker, …). Games can involve both sequential and simultaneous moves. BA 592 Lesson I.2 Formulating Games Are Moves Sequential or Simultaneous?

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5 5 Bargaining can be a Sequential Move Game. Suppose you are bargaining for more money from your current employer. Game Theory tells you to first determine two numbers: n How much salary (say, $100,000) can I get from an alternative employer? n How much (say, $120,000) am I worth to my current employer? If you take the initiative, you can play a sequential move game. You tell your employer what it will take to keep you. Then, your employer decides whether to accept or reject that offer. What salary should you tell your employer? BA 592 Lesson I.2 Formulating Games Are Moves Sequential or Simultaneous?

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6 6 Zero Sum Games A zero-sum or constant-sum game is one in which the players' interests are in direct conflict, like kicking a Penalty kick at a Goalie (at least when you cannot hurt the goalie). n The kicker’s gains are +1 if he scores, and 0 if he does not. n The Goalie gains -1 if the kicker scores, and 0 if he does not. The sum is zero. n More generally, the sum at stake can be a constant. Zero sum games are easy to solve when there are only two players. n Player A knows Player B acts to give Player A the worst possible payoff, so Player A choose the strategy that maximizes his payoff assuming the worst happens. BA 592 Lesson I.2 Formulating Games Are Players in Pure Conflict?

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7 7 Three or more players Zero-sum games with more than 2 players are complex because there is gain from cooperation of one group of players against another.Zero-sum games with more than 2 players are complex because there is gain from cooperation of one group of players against another. n What should you do in Risk if one player becomes stronger than all other players? n What should you do in a 12-man Battle Royale wrestling match against Andre the Giant (who was 7 foot, 4 inches tall and weighed 540 pounds)? BA 592 Lesson I.2 Formulating Games Are Players in Pure Conflict?

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8 8 Prisoners' Dilemma is a Non-Zero Sum Game A game is non-zero/non-constant sum if players’ interests are not always in direct conflict, so that there are opportunities for both to gain by cooperation. For example, prisoners spend less time in prison if they both choose to not confess in the following Prisoners' Dilemma. BA 592 Lesson I.2 Formulating Games Are Players in Pure Conflict?

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9 9 BA 592 Lesson I.2 Formulating Games Are Players in Pure Conflict? The Prisoners’ Dilemma Game has two suspects separately interrogated and invited to confess. One of them, say A, is told, “If the other suspect, B, does not confess, then you can cut a very good deal for yourself by confessing. But if B does confess, then you would do well to confess, too; otherwise the court will be especially tough on you. So you should confess no matter what the other does. B is told to confess, with similar reasoning. Hence, both A and B confess. But it would have been better for both if neither had confessed, because the police had no really compelling evidence against them.

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10 Other Prisoner Dilemmas (with gains from cooperation): Nuclear arms races. Dispute resolution and hiring a lawyer. Other examples? BA 592 Lesson I.2 Formulating Games Are Players in Pure Conflict?

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11 Is the Game Repeated? That is, one-shot or repeated? One-shot: play of the game occurs once.One-shot: play of the game occurs once. n Players likely to not know much about one another. n Example: tipping in a foreign country. Repeated: play of the game is repeated with the same players.Repeated: play of the game is repeated with the same players. n Reputation matters. You can gain from honesty. n Enforcement for current cooperation may arise because of the threat of future punishment. n Example: tipping in your neighborhood. Advise: If you are thinking of pursuing a dishonest or aggressive strategy, ask yourself whether you are in a one-shot or in a repeated game. If a repeated game, think again.Advise: If you are thinking of pursuing a dishonest or aggressive strategy, ask yourself whether you are in a one-shot or in a repeated game. If a repeated game, think again. BA 592 Lesson I.2 Formulating Games Is the Game Repeated?

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12 Do Players have Full or Equal Information? Chess and tic-tac-toe are games of perfect information:Chess and tic-tac-toe are games of perfect information: n Each time it is your turn, you know perfectly what your opponent has played. Setting prices in business is a game of imperfect information:Setting prices in business is a game of imperfect information: n There is strategic uncertainty when you do not know rivals’ prices. n There is external uncertainty when you do not know customer demand. Job interviews are games of asymmetric information:Job interviews are games of asymmetric information: n You know your talents and work ethic better than an employer. n You may want to signal you are smart and hard working by completing hard classes in college. n An employer may want to screen (sift or separate) for smart and hard- working job applicants by requiring the completion of a certain number of hard classes in college. BA 592 Lesson I.2 Formulating Games Do Players have Full or Equal Information?

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13 Are the Rules Fixed? In theory, all humanity has been playing one large game from the beginning of time. In practice, we analyze only subgames.In theory, all humanity has been playing one large game from the beginning of time. In practice, we analyze only subgames. Reconsider bargaining for more money from your current employer: you can get $100,000 from an alternative employer, and you are worth $120,000 to your current employer.Reconsider bargaining for more money from your current employer: you can get $100,000 from an alternative employer, and you are worth $120,000 to your current employer. If you take the initiative, you select the subgame where you move first and tell your employer what it will take to keep you, Then, your employer decides whether to accept or reject.If you take the initiative, you select the subgame where you move first and tell your employer what it will take to keep you, Then, your employer decides whether to accept or reject. If your employer takes the initiative, he selects the subgame where he tells you what he is willing to pay to keep you. Then, you decide whether to accept or reject.If your employer takes the initiative, he selects the subgame where he tells you what he is willing to pay to keep you. Then, you decide whether to accept or reject. Deciding on the subgame is called the pregame.Deciding on the subgame is called the pregame. BA 592 Lesson I.2 Formulating Games Are the Rules Fixed?

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14 Are Agreements to Cooperate Enforceable? Agreements to cooperate in a group (say, limiting production among OPEC countries) can succeed if all players act immediately and in the presence of the whole group.Agreements to cooperate in a group (say, limiting production among OPEC countries) can succeed if all players act immediately and in the presence of the whole group. Even when actions are taken in private, agreements can still be enforced by a third party if actions are observable.Even when actions are taken in private, agreements can still be enforced by a third party if actions are observable. Without observation and enforcement, agreements stand only when it is in all participants’ individual interests to comply.Without observation and enforcement, agreements stand only when it is in all participants’ individual interests to comply. Games where joint-action agreements are enforceable are called cooperative.Games where joint-action agreements are enforceable are called cooperative. Games where joint-action agreements are not enforceable are called noncooperative.Games where joint-action agreements are not enforceable are called noncooperative. Repeated interactions can allow cooperation in noncooperative games. (Like when prisoners threaten potential informants.)Repeated interactions can allow cooperation in noncooperative games. (Like when prisoners threaten potential informants.) BA 592 Lesson I.2 Formulating Games Are Agreements Enforceable?

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15 BA 592 Lesson I.1 Basic Ideas and Examples Terms and Background Assumptions Games have three parts. PlayersPlayers How many players are there?How many players are there? Does chance play a role? (Is Nature a player?)Does chance play a role? (Is Nature a player?) Strategies for each player.Strategies for each player. Payoffs are the consequences for each player for every possible profile of strategy choices for all players.Payoffs are the consequences for each player for every possible profile of strategy choices for all players.

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16 BA 592 Lesson I.2 Formulating Games Terms and Background Assumptions Strategies are the choices available to each player. If a game has purely simultaneous moves made only once, then each player’s strategy is just the action on that single occasion.If a game has purely simultaneous moves made only once, then each player’s strategy is just the action on that single occasion. If a game has sequential moves, then each player’s strategy allows actions at any moment to respond to previous actions by all players.If a game has sequential moves, then each player’s strategy allows actions at any moment to respond to previous actions by all players. If he does A, I will do X; but if he does B, I will do Y.If he does A, I will do X; but if he does B, I will do Y. For example, a check-raise in poker. I check (zero bet). If he checks, then betting is over; but if he bets, then I will raise his bet.For example, a check-raise in poker. I check (zero bet). If he checks, then betting is over; but if he bets, then I will raise his bet. A complete strategy can, in principle, be programmed into a computer, which can then handle all possible situations.A complete strategy can, in principle, be programmed into a computer, which can then handle all possible situations. A programmed poker strategy is called a bot.A programmed poker strategy is called a bot.

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17 BA 592 Lesson I.2 Formulating Games Terms and Background Assumptions Payoffs are the consequences of player strategies. Payoffs are most often monetary.Payoffs are most often monetary. Payoffs are often estimated, so predictions should be checked for robustness to varying payoffs.Payoffs are often estimated, so predictions should be checked for robustness to varying payoffs. Payoffs to one player can include another players well-being:Payoffs to one player can include another players well-being: A leader’s payoff can include a follower’s well-being.A leader’s payoff can include a follower’s well-being. A parent’s payoff can include a child’s well-being (as in the brush- your-teeth game).A parent’s payoff can include a child’s well-being (as in the brush- your-teeth game). Payoffs depending on a random variable (nature or player strategies) are evaluated according to expected value.Payoffs depending on a random variable (nature or player strategies) are evaluated according to expected value. Expected value implies risk neutrality when payoffs are monetary.Expected value implies risk neutrality when payoffs are monetary. Expected value implies risk aversion if payoffs were the utility value of money, where utility is strictly concave.Expected value implies risk aversion if payoffs were the utility value of money, where utility is strictly concave. Your Game Theory Course assumes risk neutrality.Your Game Theory Course assumes risk neutrality.

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18 BA 592 Lesson I.2 Formulating Games Terms and Background Assumptions Rationality is most often assumed: Players are perfect calculators and perfect followers of their best strategies. Players have perfect knowledge of their own interests.Players have perfect knowledge of their own interests. Players make perfect calculations what actions best serve their interests.Players make perfect calculations what actions best serve their interests. Rationality does not imply selfishness, since payoffs may include the well-being of others.Rationality does not imply selfishness, since payoffs may include the well-being of others. Rationality is unrealistic in complex games, such as chess. But players perhaps approach rational play with experience.Rationality is unrealistic in complex games, such as chess. But players perhaps approach rational play with experience. Poker professionals (sharks) may assume other professionals are rational, but do not assume bad players (fish) are rational.Poker professionals (sharks) may assume other professionals are rational, but do not assume bad players (fish) are rational. Sharks learn how to exploit mistakes by fish.Sharks learn how to exploit mistakes by fish. “If you cannot spot the fish at a table, you are the fish.”“If you cannot spot the fish at a table, you are the fish.”

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19 BA 592 Lesson I.2 Formulating Games Terms and Background Assumptions Common knowledge of rules is always assumed. Players have perfect knowledge of the rules.Players have perfect knowledge of the rules. Players assume other players have perfect knowledge of the rules.Players assume other players have perfect knowledge of the rules. Players assume other players assume other players have perfect knowledge of the rules.Players assume other players assume other players have perfect knowledge of the rules. And so on forever.And so on forever. The rules of the game has four parts:The rules of the game has four parts: The list of players.The list of players. The available strategies of each player.The available strategies of each player. The payoffs of each player given strategies chosen by all players.The payoffs of each player given strategies chosen by all players. The assumption that each player is a rational maximizer.The assumption that each player is a rational maximizer.

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20 BA 592 Lesson I.2 Formulating Games Terms and Background Assumptions Equilibrium is when each player’s strategy is a best response given beliefs about the strategies of the other players. Equilibrium in sequential-move games (like chess) can involve changing actions as the game progress (move pawn, move knight, …)Equilibrium in sequential-move games (like chess) can involve changing actions as the game progress (move pawn, move knight, …) Typical game-theory classes find equilibrium for only simple games with 2 or 3 players, each of which has 2 or 3 strategies because game-theory students cannot solve more complicated games in class.Typical game-theory classes find equilibrium for only simple games with 2 or 3 players, each of which has 2 or 3 strategies because game-theory students cannot solve more complicated games in class. Computers can help solve more complex (and more realistic) games in the workplace.Computers can help solve more complex (and more realistic) games in the workplace.

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21 BA 592 Lesson I.2 Formulating Games Review Problems Question Page 43, Question U1. Answer Some questions for each decision: What are the strategies? What are the payoffs? Do payoffs depend on the strategies chosen by other players?

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22 BA 592 Lesson I.2 Formulating Games Review Problems Question Page 43, Question U2. Answer The six dimensions of a game are: 1.Are Moves Sequential or Simultaneous? 2.Are Players in Pure Conflict? 3.Is the Game Repeated? 4.Do Players have Full or Equal Information? 5.Are the Rules Fixed? 6.Are Agreements Enforceable?

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23 BA 592 Lesson I.2 Formulating Games Review Problems Question Page 44, Question U3. Answer Think of the Prisoners’ dilemma.

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24 BA 592 Lesson I.2 Formulating Games Review Problems Question Page 44, Question U4. Answer Of the 2 3 possible permutations of the coins, how many ways can Alice win? Bob? Confucius?

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25 BA 592 Lesson I.2 Formulating Games Review Problems Question Page 44, Question U5. Answer Can there be surprises in the Penalty Kick Game?

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26 End of Lesson I.2 BA 592 Game Theory BA 592 Lesson I.2 Formulating Games

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