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Game

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any situation in which two or more individuals make choices from a menu of actions and each individual's welfare depends on the choices of others, as well as their own.

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Prisoner's Dilemma

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Two suspects arrested. Police lack evidence to convict unless one finks. If neither fink, they get a minor penalty. If one finks, he gets rewarded and the other gets maximum penalty. If they both fink, they both get a high penalty

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Stag Hunt

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Each hunter can choose to hunt hare or stag. Each prefers for the group to catch a stag, next to catch a hare, last to return empty handed. If and only if they all choose stag, they catch one. Anyone who hunts a hare catches one. Those who hunt stag while at least one other person hunts hare end up empty-handed.

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Cold War's MAD

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USA and USSR can choose a first nuclear strike or not. In a second period they can choose a second strike or not. They prefer to strike twice without the other striking. Next, they prefer to strike twice with the other side striking only once. The worst outcomes are those with two strikes from the opponent.

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Game Theory

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systematic study of the strategic interdependence of games.

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Decision Theory

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study of individual decision making when the choices of others do not affect outcomes.

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Rational Choice

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Decision maker has set of available actions that lead to different outcomes. Also has clearly defined preferences. Can rank any two outcomes. No intransitivity's. Rational is the best possible action according to preferences over outcomes.

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Why is rationality often misconstrued?

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No restrictions on the content of preferences: need not be selfish, may experience emotions, may have unconventional tastes.

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Formal Game Theory Origin

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Cournot Competitions. 19th century

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John von Neumann

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modern game theory- Theory of Games and Economic Behavior in 1944

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John F Nash Jr

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formalized Nash equilibrium- used in evolutionary biology, engineering

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Modern Game Theory

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default analytical framework in economics and political science

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Game in strategic form

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1. Set of players. 2. Set of actions available to each player. 3. Utility/Payoff Function for each player that assigns a payoff value to each outcome of the game.

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What are the two minimal assumptions we use in strategic form games?

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1. Players choose actions simultaneously, without knowing the others choice. 2. Once all players fix their choice, the outcome of the game is realized.

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Players

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individuals whose actions are pertinent in the particular strategic interaction we are trying to model.

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How do we denote the number of players?

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n >=2

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How do we denote the set of players?

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N = {1,....,n}

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How do we refer to some individual player?

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Use the index i, i in N.

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Actions/Strategies

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set of actions of player i is an exhaustive list of the relevant choices available to that player

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Inaction

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included in action set if relevant. In a game of voting- A, B, Abstain

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How do we denote a generic action of player i?

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ai

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How do we denote the set of actions of player i?

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Ai

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Outcome

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once all players choose their actions, we have all available information to determine the outcome of the game. List of action choices, one for each player in the game.

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Profile of Actions

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list of action choices, one for each player in the game. Given n sets A1....An, we take one element from each set. a = (a1,...,an)

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A-i

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Way to refer to the action of all players except i

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How could we write an entire action profile?

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a = (ai, a-i)

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How do we denote the set of all action profiles?

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A

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How do we denote the set of all action profiles by players other than i?

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A-i

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Utility

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ui: A--> R Assigns a payoff, a real number, to each outcome of the game. Represents a ranking of the outcomes of the game according to the preferences of player i. Higher ranked outcomes get higher numbers, equally ranked outcomes get the same number.

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Matrix Representation of Games in Strategic Form

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Row players payoffs appear first, followed by the payoff for the column player. Each cell represents a profile.

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Ordinal Payoff Functions

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only the resultant ranking among outcomes is relevant, not the actual size of the number assigned.

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Bach Stravinsky

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Alice and Bob both want to go different places. The worst outcome is they end up in different places.

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What does rationality assume about each player?

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Know players action sets so knows the set of possible outcomes A Has preferences over these outcomes as represented by the payoff function ui Chooses her action so that it is the best possible given these preferences

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What don't rational players need to know?

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Other players payoffs or rationality.

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Dominant Strategy

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action/strategy of player i is dominant if i gets a higher payoff using it as opposed to any other available action, irrespective of the action choices of other players.

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Actions ai is dominant for player i if

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ui(ai, a-i) > ui (a'i, a-i) for all other actions a'i in Ai and all a-i in A-i.

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Why is PD trivial?

answer

No strategic considerations. Introspection should convince them of the best course of action.

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Why is PD attention worthy?

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Socially inferior outcome ensues when reasoning is applied.

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Dominant Strategies and Rational Players

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Any rational player MUST use dominant actions/strategies.

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Strictly Dominated Action

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there exists another action that yields a strictly higher payoff for all action choices by other players.

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ai is strictly dominated if

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There exists a'i in Ai such that ui (a'i,a-i) >ui(ai,a-i) for all a-i in A-i.

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Never-A-Best-Response Property

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Strictly dominated actions cannot be a good choice for any profile of action choices by other players.

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Dominant Actions and Strict Domination

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Dominant actions are all those that strictly dominate all other available actions.

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What hinges on what player 1 knows about the rationality and payoffs of player 2?

answer

Can deduce that they will not play L, which now makes H strictly dominated.

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Iterated Elimination of Strictly Dominated Actions

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1. Identify any strictly dominated actions and delete them. 2. Repeat the process in the resultant game, until there are no more possible eliminations.

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Common Knowledge

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3 Rounds of IESDA 1. Players are rational 2. Players know that other players are rational and their payoffs 3. Players know that they know each other's payoffs and rationality. If these statements go to infinity, it is common knowledge.

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Dominance Solvable

answer

a unique profile of actions survives iterated elimination of strictly dominated actions.

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Does IESDA solution depend on the order of elimination?

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In finite dominance solvable games: no

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Weakly Dominated Action

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there exists another action that yields a strictly higher payoff for some action choices of other players and equal payoff for the remaining choices of other players.

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Action ai is weakly dominated if

answer

There exists a'i in Ai such that Ui (a'i, a-i) > = ui(ai,a-i) for all a-i in A-i, with the inequality strict for at least one a-i.

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Are strictly dominated actions weakly dominated?

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Yes, everyone. Not all weakly dominated actions are strictly dominated.

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Do players ever use weakly dominated actions?

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Yes. No reason they cannot using rationality or common knowledge.

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Does the solution of IEWDA depend on order of elimination?

answer

Yes, sometimes.

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If players' payoffs and rationality are common knowledge..

answer

players only play actions that survive iterated elimination of strictly dominated strategies

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Equilibrium

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a condition in which all influences acting upon it are canceled by others, resulting in a stable, balanced or unchanging system.

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What constitutes an equilibrium in social situations?

answer

Large population of players who repeatedly participate in the same game situation with other players of the same population. Don't play the same person twice. Can only exist if no forces exist to change individual behavior. No participant can improve by changing their own strategy.

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Social Equilibrium

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The strategy that is best for one player often depends on prevailing choices in the population at large.

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Nash Equilibrium

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profile of action choices such that no individual wishes to change her action given what everyone else is doing.

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Action profile a* in A is a Nash equilibrium if

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ui (ai*, a-i*) > = ui (ai, a-i*) for all players i and for all ai in Ai.

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What is the relationship between rationality and equilibrium?

answer

Nash equilibria are not implied from rationality or common knowledge, but the fact that we have come to expect others to behave in certain ways.

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Do Nash equilibrium prevail in every game.

answer

No.

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When is Nash equilibrium most applicable?

answer

Players have considerable experience or have learned expected behavior via observation/socialization.

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Everyday life is full of behavior patterns we expect to persist

answer

Politicians in democracies apply similar tactics across different societies.

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Multiplicity of Nash Equilibria

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explains social/cultural differences. Driving Lane Matriarchy, Patriarchy, Equality Collective Action (Stag, Hare)

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What does Nash equilibrium entail for human welfare?

answer

Nothing about the theory of Nash equilibria guarantees that Nash equilibrium outcomes are good for society. Conflict and war have roots in human incentives.

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Disequilibrium

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likely to occur when a social situation occurs infrequently and players do not understand the game/have beliefs about others play. They could also have a difficult time coordinating expectations on one equilibrium over time.

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How can Nash equilibria still be useful in disequilibrium?

answer

Informs possible behaviors that could become persistent.

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Aumann and Brandenburger

answer

Set of sufficient conditions for Nash equilibrium, premised on rationality and knowledge.

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Mutually Known

answer

If all players are rational and their profile of strategy choices a* in A is mutually known, then a* is a Nash equilibrium.

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Since i is rational and knows others' actions a*-i, it must be that ai* is such that

answer

ui (a*i, a*-i) >= ui(ai, a*-i) for all ai in Ai.

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What are sufficient conditions for Nash equilibria to prevail?

answer

1 Players are rational 2 Players expect the same outcome with probability one.

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Is rationality required in Nash equilibria

answer

No. Evolutionary biologists study ecosystems using Nash equilibria, where animal populations strategies are hard wired. Strategies that fair better are naturally selected.

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Strictly Dominated Strategies and Nash Equilibrium

answer

Can never be part of a Nash equilibrium. Suppose ai is strictly dominated by a'i and (ai, a-i) is a Nash Equilibrium. ui(a'i, a-i) > ui (ai, a-i), which cannot be true

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IESDA and Nash Equilibrium

answer

Strategies that do not survive IESDA cannot be part of Nash equilibrium. a is a Nash equilibrium but some strategy ai is eliminated in iesda. That means at that iteration, there must be some a'i where ui(a'i,a-i) > ui(ai, a-i), which cannot be true.

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If a profile is the unique survivor after iesda..

answer

it is a Nash equilibrium. Suppose that only a survives but it is not a Nash equilibrium. There exists a'i such that ui(a'i, a-i)> ui(ai, a-i). Suppose a'i is the most profitable deviation for i. Since ai does not survive iesda, it is dominated by a''. u(a'', a-i) > u(a'i, a-i) > ui(ai,a-u) --> Contradicts the hypothesis that a'i is the most profitable deviation

question

Can weakly dominated strategies be part of Nash equilibrium?

answer

Yes.

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Finding Nash Equilibrium in Small Games

answer

For each player i and each possible action choice by the other players, identify i's optimal action and underline the corresponding payoff. Combinations of actions that are mutual optimal choices of actions (all payoffs underlined) are Nash equilibria.

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Finding Nash Equilibrium in Large Games

answer

To show that a profile or group of profiles is NOT in Nash equilibrium, it suffices to identify one player that has a profitable deviation to a better action. Allows us to eliminate large groups of profiles from a set of candidate equilibria. For the remaining profiles, we need to verify no player wants to deviate.

question

n-Player Stag Hunt

answer

Players: N = {1,...,n} Actions: Ai = {Stag, Hare} Payoffs: ui(a) = 2 if aj = S for all j. 1 if ai=H, 0 if ai = S and aj =H for some j.

question

Consider all profiles a with 0 <NS(a) <n

answer

Consider i with ai=S;then ui (S, a-i) = 0

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