Game Theory

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Game Theory

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Introduction

• _{Game theory is a theoretical framework for conceiving social} situations among competing players.

• _{In some respects, game theory is the science of strategy, or at least} the optimal decision-making of independent and competing actors in a strategic setting.

• _{The key pioneers of game theory were mathematicians John von} Neumann and John Nash, as well as economist Oskar Morgenstern.

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Introduction – cont…

• _{Players act to maximize their own welfare}

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Applications

• _{Game theory has a wide range of applications, including:}

• _psychology,

• _{evolutionary biology,} • _war,

• _politics,

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• _{Game theory has been applied to analyze:}

• _Oligopolies • _{Cartels: OPEC}

• _{Tax competition across jurisdictions, countries} • _{Military strategies}

• _{Externalities: using common resources like fishery}

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Formal definition

• _Game:_{Any set of circumstances that has a result dependent on the actions of two or}

more decision-makers (players)

• _Players:_{A strategic decision-maker within the context of the game}

• _{Strategy: A complete plan of action a player will take given the set of circumstances}

that might arise within the game

• _Payoff:_{The payout a player receives from arriving at a particular outcome (The payout}

can be in any quantifiable form, from dollars to utility.)

• Information set: The information available at a given point in the game (The term information set is most usually applied when the game has a sequential component.)

• _Equilibrium:_{The point in a game where both players have made their decisions and}

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Nash Equilibrium

• _{Nash equilibrium is a concept within game theory where the optimal}

outcome of a game is where there is no incentive to deviate from their initial strategy.

• More specifically, the Nash equilibrium is a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice.

• Overall, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies.

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• _{It is a play where each strategy is a best response to the other}

• _{The Nash equilibrium is the solution to a game in which two or more} players have a strategy, and with each participant considering an opponent’s choice, he has no incentive, nothing to gain, by switching his strategy.

• _{In the Nash equilibrium, each player's strategy is optimal when} considering the decisions of other players.

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How to test Nash Equilibrium?

• _{To quickly test if the Nash equilibrium exists, reveal each player's} strategy to the other players.

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An example of Nash Equilibrium

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Cooperative game theory

• _{Cooperative game theory deals with how coalitions, or cooperative} groups, interact when only the payoffs are known.

• _{It is a game between coalitions of players rather than between} individuals, and it questions how groups form and how they allocate the payoff among players.

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Non-cooperative games

• _{Non-cooperative game theory deals with how rational economic} agents deal with each other to achieve their own goals.

• _{The most common non-cooperative game is the strategic game, in} which only the available strategies and the outcomes that result from a combination of choices are listed.

• _{A simplistic example of a real-world non-cooperative game is} Rock-Paper-Scissors.

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Symmetric Vs Asymmetric games

• _{A symmetric game is a game where the payoffs for playing a particular} strategy depend only on the other strategies employed, not on who is playing them.

• _{If the identities of the players can be changed without changing the} payoff to the strategies, then a game is symmetric.

• _{Many of the commonly studied 2×2 games are symmetric.}

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• _{Most commonly studied asymmetric games are games where there} are not identical strategy sets for both players.

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Zero-sum / Non-zero sum games

• Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources.

• _{In zero-sum games the total benefit to all players in the game, for every}

combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).

• _{Poker exemplifies a zero-sum game (ignoring the possibility of the}

house's cut), because one wins exactly the amount one's opponents lose.

• _{Other zero-sum games include matching pennies and most classical}

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Simultaneous/ Sequential games

• Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively simultaneous).

• _{Sequential games (or dynamic games) are games where later players have}

some knowledge about earlier actions.

• _{This need not be perfect information about every action of earlier players;}

it might be very little knowledge.

• _{For instance, a player may know that an earlier player did not perform one}

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The Prisoner’s dilemma

• _{The Prisoner's Dilemma is the most well-known example of game} theory. Consider the example of two criminals arrested for a crime.

• _{Prosecutors have no hard evidence to convict them.}

• _{However, to gain a confession, officials remove the prisoners from} their solitary cells and question each one in separate chambers.

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2x2 box

• _{If both confess, they will each receive a five-year prison sentence.}

• _{If Prisoner 1 confesses, but Prisoner 2 does not, Prisoner 1 will get} three years and Prisoner 2 will get nine years.

• _{If Prisoner 2 confesses, but Prisoner 1 does not, Prisoner 1 will get 10} years, and Prisoner 2 will get two years.

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• _{The most favorable strategy is to not confess.}

• _{However, neither is aware of the other's strategy and without} certainty that one will not confess, both will likely confess and receive a five-year prison sentence.

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Tit for Tat

• _{The expression "tit for tat" has been determined to be the optimal} strategy for optimizing a prisoner's dilemma.

• _{Tit for tat was introduced by Anatol Rapoport, who developed a} strategy in which each participant in an iterated prisoner's dilemma follows a course of action consistent with his opponent's previous turn.

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Dictator game

• _{This is a simple game in which Player A must decide how to split a} cash prize with Player B, who has no input into Player A’s decision.

• _{While this is not a game theory strategy per se, it does provide some} interesting insights into people’s behavior.

• _{Experiments reveal about 50% keep all the money to themselves, 5%} split it equally, and the other 45% give the other participant a smaller share.

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• _{The catch is if the second player rejects the amount offered, both A} and B get nothing.

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Volunteer’s dilemma

• _{In a volunteer’s dilemma, someone has to undertake a chore or job} for the common good.

• _{The worst possible outcome is realized if nobody volunteers.}

• _{For example, consider a company in which accounting fraud is} rampant, though top management is unaware of it.

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The Centipede game

• _{The centipede game is an extensive-form game in game theory in which two}

players alternately get a chance to take the larger share of a slowly increasing money stash.

• _{It is arranged so that if a player passes the stash to his opponent who then}

takes the stash, the player receives a smaller amount than if he had taken the pot.

• _{The centipede game concludes as soon as a player takes the stash, with that}

player getting the larger portion and the other player getting the smaller portion.

• _{The game has a pre-defined total number of rounds, which are known to}

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Limitations of game theory

• _{The biggest issue with game theory is that, like most other economic} models, it relies on the assumption that people are rational actors that are self-interested and utility-maximizing.

• _{Of course, we are social beings who do cooperate and do care about} the welfare of others, often at our own expense.

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Game theory and international

relations

• _{Iran’s nuclear research program generates global suspicion and} concern

• _{We read news about the possibility of an Israeli or a U.S. attack upon} Iranian nuclear facilities.

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• _{We can qualify the above interaction as constituting a game, that is, a} situation of strategic interdependence.

• _{Each decision maker acts in function of actions the other (or others)} can take.

• _{The best choices of Iran and the attacker depend on their forecasts of} each other’s behavior.

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• _{International conflict and other phenomena in international relations} occur as a result of decisions made by people.

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Minimax game

• _{We now consider games with two players, whom we call MAX and} MIN

• _{MAX moves first, and then they take turns moving until the game is} over.

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Conclusion

• _{In this presentation, an overview of game theory has been presented} • _{Various types of games are discussed}

• _{We discuss how Nash equilibrium provides a solution to game theory} • _{We also looked at how game theory can be used in international}