Game Theory • Our discussion comes from Bierman, H. Scott and Luis Fernandez. 1998. Game Theory with Economic Applications. 2nd ed. Addison-Wesley. • Game Theory – the study of how individuals make decisions when they are aware that their actions affect each other and when each individual takes this into account. • History: Introduced in 1944 by John von Neumann and Oskar Morgenstern in “The Theory of Games and Economic Behavior.” • The work of von Neuman and Morgenstern was expanded upon by John Nash.
Introduction to Game Theory • A game is a situation in which a decision-maker must take into account the actions of other decision-makers. Interdependency between decision-makers is the essence of a game. • In games people must make strategic decisions. Strategic decisions are decision that have implications for other people. • Game Elements 1. Set of Players. 2. Order of Play. 3. Description of the information available to any player at any point during the game. 4. Set of actions available to each player when making a decision. 5. Outcomes that result from every possible sequence of actions by the players. 6. A payoff from the outcomes. 7. Strategic situations with the above elements is considered to be well defined.
Cooperative and Non-Cooperative Games • Non-Cooperative Games are games in which players cannot enter binding agreements with each other before the play of the game. • Cooperative Games are games in which players can enter binding agreements with each other before the play of the game. • In class we only review non-cooperative games.
The Players • Players in a Game 1. Players are decision-makers. 2. Nature is a special type of player. Nature chooses actions according to fixed probabilities. 3. Strategic players are assumed to be rational decision-makers.
Three Basic Assumptions • Assumption 1: Decision-makers are rational. • Rational behavior: Decision-makers are assumed to make their choices according to internally consistent criteria. • Assumption 2: The rationality of all players is common knowledge. • Common knowledge: A ‘fact’ in a game is said to be common knowledge if every player know it, and every player knows that every other player knows it, etc. • Assumption 3: The complete description of the game (players, actions, strategies, order of play, information, and payoffs) is also common knowledge. • Time permitting, we may examine games where this last assumption is relaxed. • With these assumptions in hand, we can look at the order of play. Before we discuss this issue, we must first review decision theory.
Order of Play • Order of Play is the sequence in which decisions are made. • Sequential-move game is a game where players make decisions in a sequence, one after another like chess (which we deal with later). • Simultaneous-move game is a game where players make their decisions at the same time, such as most sports. • To understand how moves are made, we must briefly discuss the issue of information.
Information • Information – any observation or knowledge that would lead someone to reevaluate her/his probability assessments. • If information has value, the following must be true: • 1. the information must alter the decision maker’s optimal action at some decision node. • 2. the information must be revealed to the decision maker before the critical decision node is reached. • Perfect recall – A game has perfect recall if no player forgets any information she/he once knew, and all players know the actions they previously took. • Perfect information – Every player as every decision node knows the actions taken previously by every other player (including Nature). • Imperfect information – Players do not have perfect information.
Actions, Strategies, and Payoffs • Actions – The set of choices available at each decision node in a game. • Pure strategy – a rule that tells the player what action to take at each of her information sets in the game. • Mixed strategy – when players can choose randomly between the actions available to them at every information set. • Example: Play calling in sports is a mixed strategy. • Payoffs, for our purposes, consist of either profits to firms, or income to individuals. Payoffs can also be characterized in terms of utility.
Solving Games: Nash Equilibrium • Solution Concept – a methodology for predicting player behavior. • Nash Equilibrium - a collection of strategies one for each player, such that every player's strategy is optimal given that the other players use their equilibrium strategy.
Dominant and Dominated Strategies • Payoff matrix – a matrix that displays the payoffs to each player for every possible combination of strategies the players could choose. • Dominant Strategy – a strategy that is always strictly better than every other strategy for that player regardless of the strategies chosen by the other players. • Dominated Strategy – a strategy that is always strictly worse than some other strategy for that player regardless of the strategies chosen by the other players.
Weakly Dominate Strategies • Weakly dominant strategy - a strategy that is always equal to or better than every other strategy for that player regardless of the strategies chosen by the other players. • Weakly Dominated Strategy – a strategy that is always equal to or worse than some other strategy for that player regardless of the strategies chosen by the other players.
Prisoner’s Dilemma • Scenario: Two people are arrested for a crime • The elements of the game: • The players: Prisoner One, Prisoner Two • The strategies: Confess, Don’t Confess • The payoffs: • Are on the following slide (payoffs read 1,2)
Prisoner’s Dilemma, cont. Prisoner 2 Confess Don’t Confess Confess 2 years, 2 years 0 years, 10 years Prisoner 1 Don’t Confess 10 year, 0 years 5 years, 5 years • Dominant strategy equilibrium: In this game, the dominant strategy for each prisoner is to confess. So the outcome of the game is that they each get two years. • This illustrates the prisoner’s dilemma: games in which the equilibrium of the game is not the outcome the players would choose if they could perfectly cooperate.
The Advertising Game • Scenario: Two firms are determining how much to advertise. • The elements of the game: • The players: Firm 1, Firm 2 • The strategies: • High advertising, low advertising
Advertising Game, Cont. • The payoffs: Are as follows (payoffs read 1,2) Firm 2 High Low High 40,40 100, 10 Firm 1 Low 10, 100 60,60 • Dominant strategy equilibrium: In this game, the dominant strategy for firm 1 and firm 2 is high. So the outcome of the game is 40,40. • Again, this is an example of the prisoner’s dilemma. The equilibrium of the game is not the outcome the players would choose if they could cooperate.
More Prisoner Dilemmas • Industrial Organization Examples • Cruise Ship Lines and the move towards ‘glorious excess’. Royal Caribbean offers a cruise with an 18 hole miniature golf course. Princess Cruises has a ship with three lounges, a wedding chapel, and a virtual reality theater. • Owners of professional sports teams and the bidding on professional athletes. • Non-IO Examples • Politicians and spending on campaigns. • Worker effort in teams. The incentive exists to shirk, a strategy that if followed by all workers, reduces the productivity of the team. More on shirking later.
Iterated Dominant Strategies • What if a dominant strategy does not exist? • We can still solve the game by iterating towards a solution. • The solution is reached by eliminating all strategies that are strictly dominated.
Example of Iterated Dominance • Down is Firm 1, Across is Firm 2
Alternative Solution Strategies • Nash Equilibrium - a strategy combination in which no player has an incentive to change his strategy, holding constant the strategies of the other players. • Joint Profit Maximization: This is the objective of a cartel. • Cut-Throat: A strategy where one seeks to minimize the return to her/his opponent. • How does the previous game change when we change the objectives of the players? • This is one of the advantages of game theory. We do not have to assume profit maximization. We still need to be able to identify the objectives of the players.
A Lack of Dominance • Down is Player 1, Across is Player 2
A Lack of Dominance, cont. • Given these payoffs, is there a dominant or dominated strategy? • If 1 chooses A, 2 will choose C • If 1 chooses B, 2 will choose B • If 1 chooses C, 2 will choose A • Likewise • If 2 chooses A, 1 will choose A • If 2 chooses B, 1 will choose B • If 2 chooses C, 1 will choose C • Therefore, no dominant or dominated strategy exists. Is there a Nash equilibrium? • What if player 1 chose C, and player 2 chose A, is this a Nash Equilibrium? • No, if player 2 chose A, player 1 would want A. • Only when both choose B, or both happy with the choice, therefore this is a Nash equilibrium.
Mixed Strategy • Pure Strategy is a rule that tells the player what action to take at each information set in the game. • Mixed strategy allows players to choose randomly between the actions available to the player at every information set. Thus a player consists of a probability distribution over the set of pure strategies. • Examples of mixed strategy games: • Play calling in sports • To shirk or not to shirk
The Shirking Game • Scenario: A worker is hired but does not wish to work. The firm will not pay the worker if there is no work, but the firm cannot directly observe the workers effort level or output. • Players: The worker, the firm • Strategy: Work or not work, monitor or not monitor • Payoffs: Work pays $100, but the worker’s reservation wage is $40 • Worker can produce $200 in revenue, but it costs $80 to monitor
The Shirking Game, Cont. • There is no dominant strategy, or iterated dominant strategy. • There is also no clear Nash Equilibrium. In other words, no combination of actions makes both sides happy given what the other side has chosen. • There are many mixed strategies. The worker could work with probability (p) of 0.7, 0.6. 0.25, etc... The same is true for the firm. Which mixed strategy should they choose? • If the worker is most likely to shirk, the firm should monitor. Likewise, if the firm is more likely to monitor, the worker should work. In any scenario, no Nash equilibrium will be found. The key is to find a strategy that makes the opponent indifferent to his/her potential choices. • A person is indifferent when the expected return from action A equals the expected return form action B.
Solving the Shirking Game • How much should the firm monitor? • E(work) = 60p + 60(1-p) = 60 • E(shirk) = 0p + 100(1-p) = 100 - 100p • 100 - 100p = 60 • 40 = 100p • p = .40 • The worker is indifferent when the probability of monitoring is 40% and the probability of not monitoring is 60%. • How much should the worker work? • E(monitor) = 20p + -80(1-p) = 100p - 80 • E(Not monitor) = 100p + -100(1-p) = 200p - 100 • 100p -80 = 200p - 100 • 20 = 100p • p = .2 • The firm is indifferent when the probability of working is 20% and the probability of not working is 80%. • How does the cost of monitoring and the worker’s reservation wage impact behavior?
Existence of Nash Equilibrium • Every game with a finite number of players, each of whom has a finite number of pure strategies, possesses at least one Nash equilibrium, possibly in mixed strategies • Final Note: If the player’s have continuous strategies (as opposed to finite strategies) a pure strategy can be found with a reaction function.
The Football Game • Scenario: A game has come down to a final play. The 49ers are on the 2 yard line with 5 seconds to go. The current score is 20-16, with the Raiders in the lead. The 49ers have two choices, run or pass. The Raiders have two choices, defend against the run or defend against the passes. • Players: 49ers, Raiders • Strategy: Play Pass or Run, Defend Pass or Run • Payoffs: Probability of success given choices
The Football Game, cont. • There is no dominant strategy, or iterated dominant strategy. • There is also no clear Nash Equilibrium. In other words, no combination of actions makes both sides happy given what the other side has chosen. • Hence this is a mixed strategy game. • Remember, a person is indifferent when the expected return from action A equals the expected return form action B.
Solving the Football Game • Should the 49ers run or pass? • E(D run) = 70p + 20(1-p) = 20+50p • E(D pass) = 30p + 80(1-p) = 80–50p • 20 + 50p = 80 –50p • 100p = 60 • p = .60 • The Raiders are indifferent when the 49ers run 60% and pass 40% of the time. • Should the Raiders defend the run or pass? • E(run) = 30p + 70(1-p) = 70 – 40p • E(pass) = 80p + 20(1-p) = 60p +20 • 70 – 40p = 60p + 20 • 50 = 100p • p = .5 • The 49ers are indifferent when the Raiders defend the run 50% of the time.
Who will win the game? • The probability that the 49ers will win the game the Nash Equilibrium strategies are adopted equals • 0.6 * 0.5 * 30 + 0.6 * 0.5 * 70 + 0.4 * 0.5 * 80 + 0.4 * 0.5 * 20 = 50 • The 49ers have a 50% chance of winning this game when each team adopts their equilibrium strategies.
The Football Game, new payoffs. • How does changing the expected payoffs alter the probabilities that each team will take each action? • The 49ers have a very good chance of scoring if they pass, and the Raiders play run defense. • Outcome of the game: • 49ers will run with a probability of 4/7 • Raiders will play the run with a probability of 2/7
Who will win the game now? • The probability that the 49ers will win the game the Nash Equilibrium strategies are adopted equals • 4/7 * 2/7 * 40 + 3/7 * 2/7 * 90 + 4/7 * 5/7 * 70 + 3/7 * 5/7 * 50 = 61.4 • The 49ers have a 61.4% chance of winning this game when each team adopts their equilibrium strategies.
The Voting Game • Non-intuitive game theory: voting paradoxes • Scenario: Three economist need to decide how much math to require for economics majors. The options are • 1) require no math • 2) require one semester calculus • 3) require two semesters calculus • Preferences of each professor: Dr. Vaitheswaran (V) L>M>H • Dr. Berri (B) M>H>L • Dr. Wu (W) H>L>M • V is the chair of the committee, and V has the power to break any ties. Voting will be done simultaneously by secret ballot. • Naive voting: Professors ignore that it is a game and simply vote their preferences. • Outcome: V breaks the tie as the chair and the students at Coe have no math requirement.
The Voting Game, Cont. • On the left are the outcome of the game, given each possible combination of votes for B and W, and each vote for V. • The outcome in bold is the preferred outcome for V. • V has a weakly dominant strategy (L). In three instances, V’s vote would be irrelevant, therefore V would not have a preference. In every other instance, V would maximize his utility by voting (L). From this we can conclude that V will vote (L).
The Voting Game, Cont. • Voting for (L) is weakly dominated by (H) and (M), since this is the least of B’s preferences. • Therefore, B will not choose (L), and we can eliminate this option.
The Voting Game, Cont. • For W, (M) is weakly dominated by (H) and (L). Given this, W will choose (H) in every instance, so (H) is weakly dominant. • The outcome of the game then is as follows: • V will vote L • W will vote H • B will vote H • The students at Coe will thus have a high math requirement, exactly the opposite of what the chair wants.
The Good, The Bad, and the Ugly • Scenario: Three gunfighters in a gun fight. The winner gets the gold. • Players: Good is the fastest, Bad is the second fastest, and Ugly is the slowest at firing a gun. • Each gunfighter only gets one shot, if he is not killed by a faster person. The winner gets the gold. If two people survive, the two agree to split the gold. • All three gunfighters know the skill level of their opponents. • Potential Actions: Shoot at one of the remaining players.
The Good, The Bad, and the Ugly, cont. • Ugly has a dominant strategy. If Ugly aims at Good, he is always better off than when he aims at Bad. • Bad has the same dominant strategy. Aiming at Good results in a higher payoff than aiming at Ugly. • Hence, in this game, the fastest gunfighter is killed.