1. GAMES
an introduction
John Bradford, Ph.D.

2. 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.

3. What are ‘Games?’
• Ingredients of a game:
1. The Players
2. Options (i.e. their options or
possible ‘moves’)
3. Outcomes
4. ‘Payoffs’ – the reward or
loss a player experiences

4. I. DESCRIBING GAMES

5. Describing Games
• We can describe ‘games’ in
three ways:
1. Verbally
2. Using a matrix (= table)
3. Using a Tree diagram

6. 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).

7. 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

8. 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

9. 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

10. 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?
Employee
0,0
Boss
2, -2
-1, 0

11. II. DOMINANT STRATEGY

12. 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?

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

14. 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

15. 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’)

16. ZERO-SUM AND VARIABLE-SUM
GAMES

17. 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

18. 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.

19. 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, -1
Matchmaker
Variety-Seeker

20. 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!

21. 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

22. Tree Diagrams
(tic-tac-toe)