EC364: Game Theory in Economics, Business & Finance

Subgame

a regular subtree together with the corresponding payoffs.

Self-enforcing agreement

agreement that is a Nash equilibrium in a game.

Rational expectations

expectations that are correct on average (stochastic fulfilment of expectations).

Nash equilibrium

fixed point where the best response of one player is mapped into the best responses of the other players.

Signalling game

game where one player sends signals to other players in order to influence their move, e.g. a job candidate suggest to the potential employer what wage is ap-propriate for him given his education (whisky vs. tonic game).

Iterated game

game which is played several times (repeated game) and that allows for a strategy to be contingent on past moves and thus for reputation effects and retribution.

Focal point

mechanism to choose between alternative Nash equilibria, introduced by Thomas Schelling in 1960 ('Strategy of Conflict').

Minimax

minimising maximum loss.

Prisoners' dilemma

non-cooperative game, where the dominant strategy of each player leads to a Pareto inefficient outcome.

Best response

optimal move (in terms of individual preferences) in response to a possible move of opponent; most important aspect of Nash equilibrium.

Risk aversion

player is prepared to pay less than the expected gain to enter the game (prefers sure thing to a gamble with the same expected value).

Tragedy of the commons

title of a famous article by Garrett Harding (1968). Problem of common (or open-access) resources provoking free-rider behaviour which is a variant of the prisoners' dilemma; there exists no excludability to use the resource; the result is tragic because all loose.

Complete preferences

when a person can compare any two options (or outcomes) to say which one is preferred or whether they are equally liked.

Credible commitment

when it is known that to stick to the commitment is a dominant strategy when it due; i.e. when it is in the own interest of the person who announces it to fulfil the commitment (problem of time inconsistency).

Four elements of strategic games:

1. players (strategies and preferences); 2. rules; 3. information; 4. payoffs.

Weakly dominant strategy

A strategy that is not dominated and dominates all other strategies for some strategies profiles of all other players.

Subjective expected utility

Expected utility with subjective probabilities.

Subgame perfect Nash equilibrium

Nash equilibrium corresponding to a profile of subgame perfect strategies.

Strategic (normal) form

Payoff matrix; extensive form - Game tree.

Time (or dynamic) inconsistency

Phenomenon that occurs when it is not in the best interest of a player to carry out a threat or promise that was initially designed to influence the other player's actions.

Sure-thing principle

Pivotal axiom in Leonard Savage's (1954) subjective probability theory. It demands a decision maker to neglect potential events for which the outcome is equal across options.

Information set

Set of decision nodes at a given stage in a game between which a player is unable to distinguish.

Subgame perfect strategy

Strategy, the relevant parts of which are best responses of a player in every subgame of the game.

Regular subtree

Subtree of an extensive form game that fulfils the following criterion: If a decision node belongs to the subtree then all decision nodes which are in the same information set belong to the subtree. Note that the whole game tree is always a regular subtree of itself.

Bayes-Nash equilibrium

Such a game involves a move of nature determining what type players are. A Bayes-Nash (or simply 'Bayesian') equilibrium is a strategy profile that prescribes optimal behaviour for each and every type of a player, given the other player's strategies and given beliefs about other player's types.

Ellsberg's paradox

a paradox in decision theory in which people's choices violate the 'sure-thing' principle of Savage's subjective expected utility theory. (Urns with red and black balls).

Subtree of an extensive form game

all possible paths a game can take starting from a particular decision node. Note that the whole game tree is a subtree of itself.

Rationality

behavioural assumption, where players have consistent preferences and expectations and act according to their best interest; a strong version of rationality assumes that actors know everything that can be known and make no mistakes.

Battle of sexes

coordination game with at least two equilibria, none of them Pareto dominating due to the different preferences (meeting at the party or at the pub).

Chicken game

coordination game with multiple Nash equilibria, but each player prefers a different equilibrium outcome. Players have options to surrender or challenge the other. Who stays longest - wins, 'chicken' - backs down.

Decision matrix

description of a game played against 'nature' (decision theory);payoff matrix - description of a game played against other actors (game theory).

Chain-store paradox

finitely repeated entry deterrence game. Using backward induction it follows that in every round, the weak monopolist has no incentive to fight, despite the number of repeats. If asymmetric information prevails, a weak monopolist should randomize between fighting and conceding while a strong monopolist should always fight.

Reaction function

function that shows the best response of one firm to any action taken by its rival; special case of best response function in terms of Cournot's duopoly game.

Sequential game

game in which players make decisions in a certain predefined order, and in which at least some players can observe the moves of players who preceded them. This type of games is represented by game trees (extensive form) and solved using the concept of backward induction and (if required) additional concepts such as subgame perfect equilibrium.

Continuous strategies game

game in which the strategy set of at least one player is infinite (example: Cournot's duopoly model).

Coordination game

game with at least two equilibria. In a pure coordination game, there is no conflict of interest and players are indifferent between equilibria. In order to realize one of those equilibria, players have to coordinate their strategies.

St. Petersburg paradox

game with infinite expected value, but actually people are willing to pay only a very small amount to play it.

Game tree

graphical representation of a sequential game (extensive form), which provides information about the players, payoffs, strategies, and the order of moves. It consists of nodes, which are points at which players take actions, connected by edges, which represent the actions that may be taken at that node. Terminal nodes represent an end to the game where the payoffs earned by each player are shown.

Maximin

maximising minimum profits.

Cournot's duopoly model

model where two firms compete in the same market, in which each firm selects the quantity that maximizes its profit in response to any (expected) output decision of the opponent. The so derived reaction function is a forerunner of the best response functions, which intersect at a fixed point forming a Nash equilibri-um.

Risk neutrality

player is indifferent between a sure thing and a gamble with the same expected value.

Risk loving

player is prepared to pay more than the expected gain to enter the game (prefer a gamble to a sure thing with the same expected value).

First mover advantage

player that moves first in a sequential game can determine the outcome of the game.

(Non-)Cooperative game - cooperative

players are able to make binding commitments (possibly via third parties). - non-cooperative: players cannot make binding commitments; hence they will act in their personal rather than collective interests (e.g. PD-games).

Bayes' rule

principle for calculating conditional probabilities. It starts from the notion that the (unconditional) probability of a joint even XY is the conditional probability for X given that Y has actually occurred times the (unconditional) probability of Y.

Expected value (EV)

probability weighted average of the payoffs corresponding to alternative outcomes of an action.

Expected utility (EU)

probability weighted average of the utilities of payoffs corresponding to alternative outcomes of an action.

Strategy

pure: a tuple of definite moves/actions a player assigns to each of his information sets; mixed: a probability distribution defined over the pure strategy of a player. A player would only use a mixed strategy when she is indifferent between pure strategies.

Move of nature

refers to external effects and random outcomes together with the distribution of probabilities of different scenarios.

Lottery

representation of a random game specifying by vector of possible outcomes with prescribed probabilities.

Selfish, altruistic and hostile preferences

selfish preferences do not take into account the outcome of one's actions on other persons; altruistic or hostile preferences take such effects into account, positive in the first, negative in the second case.

Game of trust

sequential game in extensive form, where trust is placed or not placed and the trustee may have a temptation to abuse; one-sided prisoners' dilemma (David Hume's problem).

Entry deterrence game

sequential game with two firms: A considers entering the market, B can fight or back down in response (see also chain-store paradox).

Restaurant-bill game

shows that a shared bill in a restaurant tends to be higher than if each person pays one's own bill; version of the 'tragedy of the commons'.

Backward induction

solution concept for games in extensive forms and with finite time horizon that starts from the last decision node and works backwards to the first one.

Folk theorems

state that in any infinitely repeated game an equilibrium exists that gives each player an average payoff that is not lower than their maximin payoff in the stage-game provided players are sufficiently patient. They are called 'folk theorems' be-cause nobody knows who first invented them into game theory.

Tit-for-Tat ('this for that')

strategy in an infinitely repeated prisoners' dilemma game: cooperate until partner defects, then defect, but when partner shows readiness to go back to cooperation, also change back to cooperation.

Dominant strategy

strategy that gives the highest payoff regardless an opponent's move.

Dominated strategy

strategy that regardless of opponent's move give lower payoff than another strategy, so that no rational player would choose it.

Constant-sum game

sum of all payoffs is always constant, regardless of the combination of strategies. If the payoff sum is positive, all players can win, although the distribution will generally be unequal. A special case is a zero-sum game , where one player's gain is the loss of the others.