Hon 410 Game theory; Exam 2

Bayesian game

A game in which players have private types drawn from a commonly known prior.

Type-contingent strategy

A plan naming an action for each of a player’s possible types.

Bayesian Nash equilibrium

Strategies such that every type of every player maximizes its expected payoff, given the others’ strategies and its beliefs about their types.

Threshold (cutoff) strategy

Act if your private number is on one side of a cutoff, not if it is on the other.

Signal

A costly action whose cost differs across types, so that taking it is evidence of the type.

Separating equilibrium

Different types choose different actions; the action reveals the type.

Pooling equilibrium

All types choose the same action; the action reveals nothing, and beliefs stay at the prior.

Semiseparating equilibrium

One type mixes, so the action reveals the type only partly.

Off-path belief

What a player would believe after an action nobody takes in equilibrium.

Single crossing

The signal is cheaper for the type that wants to be identified, which makes separation possible.

Least-cost separating equilibrium

The separating equilibrium with the smallest signal that still deters the other type.

Perfect / imperfect information

Whether, at every turn, a player knows every move made so far.

Complete / incomplete information

Whether every player knows every player’s payoffs (types).

Strategic / external uncertainty

Not knowing another player’s move, or not knowing Nature’s move.

Asymmetric information

One player knows something another does not.

Information set

Nodes of one player that she cannot tell apart; all have the same actions. Drawn as a dashed oval.

Nature

A non-strategic mover that chooses at random with known probabilities.

Type

A player’s private characteristics, such as payoffs, costs, or preferences. Drawn by Nature in Harsanyi’s approach.

Strategy (with types)

A plan naming an action for every information set, and so for every type.

Prior / updated (posterior) belief

The probability before and after seeing evidence.

Bayes’ rule

Updated belief = cases of that kind producing the evidence ÷ all cases producing the evidence.

Base-rate neglect

Ignoring the prior when updating, for example by treating a test’s hit rate as the answer.

Cheap talk

A costless, non-binding message. Believable when interests are aligned; ignored when they conflict.

Babbling equilibrium

An equilibrium in which messages carry no information and are ignored. It always exists

Temptation, reward, punishment, and sucker’s payoffs. A prisoners’ dilemma has T > R > P > S.

Finitely repeated game

A game played a known number of times. A prisoners’ dilemma repeated this way unravels to defection in every round.

Indefinitely repeated game

A game that continues after each round with some probability, so there is no known last round.

Discount factor (δ)

The weight on a payoff one round ahead: the continuation probability, divided by (1 + interest rate) if the future is also discounted.

Trigger strategy

A strategy that cooperates until the other defects and then punishes. Grim trigger punishes forever.

Cooperation threshold

The smallest δ at which cooperation can be sustained. For grim trigger, (T − R) / (T − P).

Tit for Tat

Cooperate first, then copy the other player’s last move.

Folk theorem

When the future counts enough, many outcomes of a repeated game can be sustained in equilibrium.

Nice / retaliatory / forgiving / clear

Axelrod’s four properties of rules that did well in his tournaments.

Noise (mistakes)

Actions that come out differently from what was intended, seen by the other player as real choices.

Echo

The alternating retaliation two Tit-for-Tat players fall into after one mistake.

Reputation

A record of past play that makes a player’s future behaviour predictable, and so worth protecting.

Strategic form of a sequential game

The tree written as a table: each player’s complete strategies as rows and columns, each cell found by playing the two strategies through the tree.

Product rule

A player’s number of strategies is the product, over that player’s nodes, of the number of actions at each node.

Noncredible threat (or promise)

A planned action at a node that the player would not choose if the node were actually reached.

Subgame

A part of a tree that starts at a single decision node and includes everything that can follow from it.

Subgame-perfect equilibrium (SPE)

Strategies that form a Nash equilibrium of the whole game and a Nash equilibrium in every subgame, reached or not. In games of perfect information, the rollback equilibrium.

Two-stage game

A game with a sequential first stage and a later stage (often simultaneous); solved by finding the equilibrium of the last stage first.

Strategic move

An action taken before the game proper to change the other player’s beliefs about how you will play.

Commitment (unconditional move)

“I will do X whatever you do,” made stick in advance.

Threat

A response rule that hurts the other player if she does what you do not want.

Promise

A response rule that rewards the other player if she does what you want.

Deterrence / compellence

A move meant to stop the other player from acting / to make her act.

Credibility

A strategic move is credible if, at the node where it would be carried out, it is the player’s best choice.

Commitment device

Something done in advance to make a move credible: reducing your freedom of action, or changing your payoffs.