This document provides an introduction to game theory and how to describe games using matrices and tree diagrams. It defines what constitutes a game, including the key elements of players, their options/moves, possible outcomes, and payoffs. Games can be zero-sum, constant-sum, or variable-sum depending on whether the total payoffs equal zero, remain constant, or vary. Matrix tables are used to describe games like Rock-Paper-Scissors and Matching Pennies by listing the options for each player and their payoffs. Tree diagrams depict games involving sequential moves rather than simultaneous choices. The concept of a dominant strategy is also introduced.

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

2. What are ‘Games?’
• In “game theory,” a ‘GAME’ is an
interaction of decision-makers.
– The Key idea is that players make
decisions that affect one another.
• Ingredients of a game:
1. The players
2. Their options (i.e. possible ‘moves’)
3. Possible outcomes
4. ‘Payoffs’- (i.e. players preferences
among those outcomes)

3. What are ‘Games?’
• A ‘Game’ in this sense includes
games like chess, tic-tac-toe,
football, basketball, etc.
• A ‘game’ as defined also
includes any real-life situation
in which our decisions
influence one another.
– (Remember the definition of
sociology?)

4. Describing Games
• Remember, a game is simply a
situation of interactive
decisions. We can describe
these interactive situations:
1. Verbally
2. Using a matrix (= table)
3. Using a Tree diagram

5. 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

6. Matrix Descriptions
Rock, Paper, Scissors
STEP 2: Write down the ‘payoffs’ (i.e. preferences) for
each possible joint outcome.
– Below I use numbers, +1 to indicate a win, -1, to
indicate a loss, and 0 to indicate a draw.
– Note that there are two different payoffs!
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

7. 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.
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

8. 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

9. Zero-sum
• In a zero-sum game, one person’s
gain comes at the expense of
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.

10. Zero-sum
• Rule: a game is zero-sum if payoffs sum to
ZERO under all circumstances.
– For example, Player 1 chooses Rock and Player 2
chooses Scissors. The aggregate payoff is:
1 – 1 = 0.
ROCK PAPER SCISSORS
ROCK 0,0 -1, +1
+1, -1
PAPER +1, -1 0, 0 -1, +1
SCISSORS -1, +1 +1, -1 0,0

11. 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

12. Constant-sum and Variable-sum
• Not all games are zero-sum games!
1. A situation in which the total payoffs are fixed
and never change, but do not necessarily equal
zero, is called a constant-sum game.
– Note: Zero-sum games are a kind of constant-sum
game in which the constant-sum is zero.
2. Variable-sum games are those in which the sum
of all payoffs changes depending on the choices
of the players! The game prisoner’s dilemma is
a classic example of this! (You will have to show
this yourself)

13. Tree Diagrams
• Tree diagrams (aka ‘decision-trees’) 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

14. Tree Diagrams
(tic-tac-toe)

15. Dominant Strategy
• In Game Theory, a player’s dominant strategy
is the 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?