GAMESan introduction

John Bradford, Ph.D.

What are ‘Games?’

• Game Theory = the study of interactive, strategic decision making among rational individuals.– A ‘GAME’ in this sense is any

form of strategic interaction!– The Key idea is that players

make decisions that affect one another.

What are ‘Games?’

• Ingredients of a game:1. The Players2. Options (i.e. their options or

possible ‘moves’)

3. Outcomes4. ‘Payoffs’ – the reward or

loss a player experiences

I. DESCRIBING GAMES

Describing Games

• We can describe ‘games’ in three ways:

1. Verbally2. Using a matrix (= table)3. Using a Tree diagram

Describing Games

1. A MATRIX (table) most easily describes a simultaneous game (where players move at the same time, like the game ‘rock, paper, scissors’)

– Note, however, that a matrix can also describe a sequential game; it’s just a little more complicated.

2. A DECISION-TREE is used to describe a sequential game (where players take turns).

Matrix Descriptions Rock, Paper, Scissors

STEP 1: Write down the options for both players in a table.– Player 1 = row chooser– Player 2 = column chooser

ROCK PAPER SCISSORS

ROCK

PAPER

SCISSORS

Matrix Descriptions Rock, Paper, Scissors

STEP 2: Write down the ‘payoffs’ (i.e. preferences) for each possible joint outcome.– Note that there are two different payoffs!

ROCK PAPER SCISSORS

ROCK tie, tie lose, win win, lose

PAPER Win, lose tie, tie lose, win

SCISSORS lose, win win, lose tie, tie

PLAYER 1

PLAYER 2

Matrix Descriptions Rock, Paper, Scissors

• By convention, the first number is the payoff to Player 1 (the row chooser). The second number is the payoff to Player 2 (the column chooser).– If you only see one number, it is always from the point of view of Player 1.– Below I use numbers, +1 to indicate a win, -1, to indicate a loss, and 0 to

indicate a draw.

ROCK PAPER SCISSORS

ROCK 0,0 -1, +1 +1, -1

PAPER +1, -1 0, 0 -1, +1

SCISSORS -1, +1 +1, -1 0,0

PLAYER 1

PLAYER 2

Decision-trees

• Decision-trees (aka tree diagrams) are useful depictions of situations involving sequential turn-taking rather than simultaneous moves.

• Asking Boss for a Raise?

Employee0,0

Boss

2, -2

-1, 0

Not ask

AskGive

NotGive

II. DOMINANT STRATEGY

Dominant Strategy

• In Game Theory, a player’s dominant strategy is a choice that always leads to a higher payoff, regardless of what the other player(s) choose.– Not all games have a dominant strategy, and games

may exist in which one player has a dominant strategy but not the other.

– In the game prisoner’s dilemma, both players have a dominant strategy. Can you determine which choice dominates the others?

IV. PRISONER’S DILEMMA (AGAIN) AND OTHER SIMPLE GAMES

PRISONER’S DILEMMA

• Remember the prisoner’s dilemma game?

• It’s basic structure is this:

COOPERATE DEFECT

COOPERATE SECOND, SECOND

WORST, BEST

DEFECT BEST, WORST

THIRD, THIRD

PRISONER’S DILEMMA• The ‘Prisoners Dilemma’ describes many

real-life situations:– Cleaning dorm rooms: best thing for you is

other guy to tidy up; but worst outcome is to tidy up for other person. What do you do?

– Economics: firms competing, driving prices low.

– Nuclear arms race– Pollution (‘Tragedy of the Commons’)

ZERO-SUM AND VARIABLE-SUM GAMES

Matrix Descriptions

• Notice that:1. Players make their moves simultaneously ( they

do not take turns), and also that,2. R…P…S… is depicted as a ZERO-SUM GAME. – “Zero-sum” refers to a situation in which the

gains of one player are exactly offset by the losses of another player. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero.• TOTAL GAINS = TOTAL LOSSES

Zero-sum

• In a zero-sum game, one person’s gain is another person’s loss.

• Example: Imagine a pizza of fixed quantity. If you eat one more slice than I do, I necessarily eat one slice less! More for you = Less for me.

• Example: A thief becomes richer by stealing from others, but the total amount of wealth remains the same.

Zero-sum• Example: ‘Matching Pennies’

– Rules: In this two-person game, each player takes a penny and places it either heads-up or tails-up and covers it so the other player cannot see it. Both players’ pennies are then uncovered simultaneously. Player 1 is called Matchmaker and gets both pennies if they show the same face (heads or tails). Player 2 is called Variety-seeker and gets both pennies if they show opposite faces (one heads, the other tails).

HEADS TAILS

HEADS +1, -1 -1, +1

TAILS -1, +1 +1, -1Matchmaker

Variety-Seeker

Variable Sum Games

• Not all life situations) are zero-sum games! • Variable-sum games are those in which the

sum of all payoffs changes depending on the choices of the players!

Variable Sum Games

– Question: is the Prisoner’s dilemma game below zero-sum or variable sum?

CONFESS NOT CONFESS

CONFESS 5 YRS, 5 YRS 0 YRS, 10 YRS

NOT CONFESS

10 YRS, 0 YR 1 YR, 1 YR

Tree Diagrams(tic-tac-toe)