SCIT1003 Chapter 3 : Prisoner’s Dilemma Non-Zero Sum Game. Prof. Tsang. Zero-Sum Games. The sum of the payoffs remains constant during the course of the game. Two sides in conflict, e.g. chess, sports Being well informed always helps a player. Example of zero-sum game. Matching Pennies.

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SCIT1003 Chapter 3 : Prisoner’s Dilemma Non-Zero Sum Game

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Zero-Sum Games • The sum of the payoffs remains constant during the course of the game. • Two sides in conflict, e.g. chess, sports • Being well informed always helps a player

Games of Conflict • Two sides competing against each other • Characteristics of zero-sum games: your loss is my gain • Simultaneous moves: lack of information about the opponent’s move • Logical circle of reasoning: I think that he thinks that I think that …

Non-Zero Sum GamePrisoner’s Dilemma • A zero-sum game is one in which the players' interests are in direct conflict, e.g. in football, one team wins and the other loses; payoffs sum to zero. • A game is non-zero-sum, if players interests are not always in direct conflict, so that there are opportunities for both to gain, e.g. games in economics • For example, when both players choose Don't Confess in the Prisoners' Dilemma • Most game in reality have aspects of common interests as well as conflict.

Imperfect Information • Partial or no information concerning the opponent is given in advance to the player’s decision, e.g. Prisoner’s Dilemma. • Imperfect information may be diminished over time if the same game with the same opponent is played repeatedly.

Games of Co-operation Players may improve payoff through • communicating • forming binding coalitions & agreements • do not apply to zero-sum games Prisoner’s Dilemma with Cooperation

Strategies • A strategy is a “complete plan of action” that fully determines the player's behavior, a decision rule or set of instructions about which actions a player should take following all possible histories up to that stage. • The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). • A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.

Dominant or dominated strategy • A strategy S for a player A is dominantif it is always the best strategies for player A no matter what strategies other players will take. • A strategy S for a player A is dominatedif there is at least a strategy better than it no matter what strategies other players will take.

Use strategy 1 Rule: If you have a dominantstrategy, use it! Opponent Strategy 1 Strategy 2 Strategy 1 150 1000 You Strategy 2 25 - 10

Dominance Solvable • If each player has a dominant strategy, the game is dominance solvable COMMANDMENT If you have a dominant strategy, use it. Expect your opponent to use his/her dominant strategy if he/she has one.

Only one player has a Dominant Strategy • For The Economist: • G dominant, S dominated • Dominated Strategy: • There exists another strategy which always does better regardless of opponents’ actions

How to recognize a Dominant Strategy To determine if the row player has any dominant strategy • Underline the maximum payoff in each column • If the underlined numbers all appear in a row, then it is the dominant strategy for the row player No dominant strategy for the row player in this example.

How to recognize a Dominant Strategy To determine if the column player has any dominant strategy • Underline the maximum payoff in each row • If the underlined numbers all appear in a column, then it is the dominant strategy for the column player There is a dominant strategy for the column player in this example.

If there is no dominant strategy • Does any player have a dominant strategy? • If there is none, ask “Does any player have a dominated strategy?” • If yes, then • Eliminate the dominated strategies • Reduce the normal-form game • Iterate the above procedure

Successive Elimination of Dominated Strategies • If a strategy is dominated, eliminate it • The size and complexity of the game is reduced • Eliminate any dominated strategies from the reduced game • Continue doing so successively

Example: Two competing Bars • Two bars (bar 1, bar 2) compete each other • Can charge price of $2, $4, or $5 for a drink • 6000 tourists pick a bar randomly • 4000 natives select the lowest price bar No dominant strategy for the both players. Bar 2

Equilibrium • The interaction of all players' strategies results in an outcome that we call "equilibrium." • Traditional applications of game theory attempt to find equilibria in games. • In an equilibrium, each player is playing the strategy that is a "best response" to the strategies of the other players. No one is likely to change his strategy given the strategic choices of the others. • Equilibrium is not: • The best possible outcome. Equilibrium in the one-shot prisoners' dilemma is for both players to confess. • A situation where players always choose the same action. Sometimes equilibrium will involve changing action choices (known as a mixed strategy equilibrium).

Definition: Nash Equilibrium “If there is a set of strategies with the property that no player can benefit by changing his/her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.” Source: http://www.lebow.drexel.edu/economics/mccain/game/game.html

Nash equilibrium • If each player has chosen a strategy and no player can benefit by changing his/her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

Conditions for Nash equilibrium • Each player is choosing a best response to what he believes the other players will do. • Each player’s beliefs are correct. The other players are all doing what everyone else thinks they are doing. Assumptions: Rational players “Putting yourself in the other person’s shoes”

Prisoner’s Dilemma : Applications • Relevant to: • Nuclear arms races. • Dispute Resolution and the decision to hire a lawyer. • Corruption/political contributions between contractors and politicians. • How do players escape this dilemma? • Play repeatedly • Find a way to ‘guarantee’ cooperation • Change payoff structure

Sustainability of resources sharing • Community resources sharing is generally viewed as a form of cooperative game similar to Prisoner’s Dilemma by most people. • However, its consequence is much deeper than the simple (& superficial) payoff matrix would suggest.

Tragedy of the Commons • When individuals, acting independently & rationally, will deplete a shared common resource even when doing so is not in their best interest. • An example to explain overuse of shared resources. • Extend the Prisoner’s Dilemma to more than two players. • Each member of a group of neighboring farmers prefers to let his cow/sheep to graze on the commons, rather than keeping it on his own inadequate land, but the commons will be rendered unsuitable for grazing if it is overgrazed.