Econ 171 Final

Game Definition

2+ players, strategic interaction

Strategic interaction

different outcomes determined by strategic action

Economic Games

Imperfect Competition, Auctions, Negotiations

Three Elements of Game

Players, Strategies, Payoffs

Non-zero sum game

an interaction in which both participants can win (or lose)

Investment Game

Sequential game, P1 chooses to invest "send" money, P2 chooses to share reward

Aggressive Strategy is bad for __ games

repeated

Strategies definition

Complete directions for what to do in any situation

Trigger Strategies

Action is taken contingent on your opponent's past decisions (repeated games)

Tit for tat

default nice, mean for 1 turn if opponent is mean

Grim Trigger

default nice, mean forever if opponent is mean

Game theory Assumption 1

Known and Fixed payoffs (acting as if expected payoff = certain payoff)

Game theory Assumption 2

Rational Behavior (players maximize payoffs)

Game theory Assumption 3

Common Knowledge (payoff, strategy, players, known to all)

Assurance Game

A game where each player has two strategies, say, Cooperate and Not, such that the best response of each is to Cooperate if the other cooperates, Not if not, and the outcome from (Cooperate, Cooperate) is better for both than the outcome of (Not, Not).

Battle of the Sexes

both player choosing same benefits both, even if payoffs different - nobody gets anything from disagreement

best-response analysis

Finding the Nash equilibria of a game by calculating the best-response functions or curves of each player and solving them simultaneously for the strategies of all the players.

chicken game

A game where each player has two strategies, say Tough and Weak, such that [1] both (Tough, Weak) and (Weak, Tough) are Nash equilibria, [2] of the two, each prefers the outcome where she plays Tough and the other plays Weak, and [3] the outcome (Tough, Tough) is worst for both.

convergence of expectations

A situation where the players in a noncooperative game can develop a common understanding of the strategies they expect will be chosen.

coordination game

multiple nash eq, 2 players negotiate to best eq

dominance solvable

A game where iterated elimination of dominated strategies leaves a unique outcome, or just one strategy for each player.

dominant strategy

strategy that works no matter what other players do

dominated strategy

strategy that fails no matter what other players do

focal point

A configuration of strategies for the players in a game, which emerges as the outcome because of the convergence of the players' expectations on it.

game table / payoff matrix / payoff table

strategy vs players axis = payoff chart

superdominant vs weakly dominant

always beat opps vs always at least equal to opps

strategic / normal form

player + strategy on both axis = payoff chart

mixed strategy

A mixed strategy for a player consists of a random choice, to be made with specified probabilities, from his originally specified pure strategies.

Nash eq

neither player can make a move to improve their payoff

ordinal payoff

Each player's ranking of the possible outcomes in a game.

pure coordination game

all nash points have same payoff for everyone (only task is to reach nash)

pure strategy

rule for player that has concrete action for any situation

successive elimination of dominated strategies

same as iterated elimination of dominated strategies

effective rate of return

Rate of return corrected for the probability of noncontinuation of an investment to the next period.

discount factor

Present value of a $1 future payment (if discount rate is 10%, discount factor is .91)

First mover advantage example

senate ads

Second mover advantage example

bidding

Limitations of Backwards Induction

Complex Games, Irrational Concerns (fairness)

Game of Perfect Information

Full history of moves/decisions are known; any point has only one "node"

Kuhn's Theorem

Every finite game of perfect information has a backward induction solution.

Kuhn's Theorem Corollary

As long as no terminal nodes are tied, then only 1 unique solution

Node

Discrete point in game tree

Playing against nature

Ignorance of nature outcome = use expected payoffs w/ probabilities

How to find # of strategies

actions per node ^ nodes

rollback equilibrium

the strategies (complete plans of action) for each player that remain after rollback analysis has been used to prune all the branches that can be pruned

winning strategy for 2 jar game

Make jars unequal, then mirror opponent moves

minimax strategy

in a zero-sum game, choosing strat that hurts opp (pure strategy)

What is Focal Point based on?

simplicity, symmetry, convention, logic (admit to crime if you did it)

evolutionary game

A situation where the strategy of each player in a population is fixed genetically, and strategies that yield higher payoffs in random matches with others from the same population reproduce faster than those with lower payoffs.

imperfect information

no knowing whole decision tree leading up to ur node

screening

strategy of less informed player to get credible info from a more informed player

strategic uncertainty

A player's uncertainty about an opponent's moves made in the past or made at the same time as her own.

screening example

asking guy to give up his apartment to to test if he's committed (with no intention of him actually doing it)

Every Nash equilibrium is a rationalizable equilibriuim

True

every rationalizable equilibrium is a Nash equilibrium

False

expected payoff

The probability-weighted average (statistical mean or expectation) of the payoffs of one player in a game, corresponding to all possible realizations of a random choice of nature or mixed strategies of the players.

opponent's indifference property

An equilibrium mixed strategy of one player in a two person game has to be such that the other player is indifferent among all the pure strategies that are actually used in her mixture.

How to find which Eq is risk dominant

Find respective gains from deviations, find product of these two values. Risk dominant one is the smaller one.

First mover advantage for infinite bargaining games

surplus(M)/1+discount factor

Mixed Strategy Nash Equilibrium

Play so opponent has indifference, solve your probabilities based on what gives them no best move

Pooling Outcomes

Separating game negative outcome, where separation mechanism fails and people of different groups make same decisions. Usually miscalculation on reward mechanisms

Screening Range

Finding the range of costs that bar one group but not the other (too low or too high leads to pooling outcomes)

If something is NASH but not subgame perfect it’s because

A threat is not credible

Subgame Perfect

Always gives highest payout no matter what other player does

Strategic form

the grid

Best Response Method (BRM)

Profit Max Eq for both firms, FOC to get BRM, plug BRM into each other

Common Value Auction, asset value

sum of player’s signals

Effective discount rate (R) - in games with unpredictable endings

discount rate + chance of continuance (if r = .10 and chance of continuance per game is .5, R = .6)

Why is pooling not equilibrium usually

the first firm who does separating successfully will profit big

Perfect complete information game

players know all move history and payoff matrix

Imperfect complete information game

simultaneous moves, know payoff matrix

Incomplete information game

Bayesian, one party doesn’t know something about the other

Bayes Nash Eq (zero-sum)

use probabilities to make other player indifferent between options

Bayes Nash Eq (lopsided rewards)

Alice and Bob, average based on possible states of world

First Price Auction best strategy (2 players)

bid v/2

First Price Auction best strategy (n players)

bid v(n-1)/n

optimal bid common value auction

ms

m equation (common value auction)

m > v / (v + x) (x = statistical estimate for highest error term by other player)

Stable Matching Game / Deferred Acceptance Algorithm

Two groups, one group proposes to other, other decides among suitors, unmatched keep playing until all are matched

Deferred Acceptance Algorithm winners

all strategic benefit flows to the side that makes the proposals.