1. Games
2. Strategies

C1 
C2 
C3 
R1 
10, 3 
7, 10 
5, 7 
R2 
20, 7 
15, 12 
10, 6 
R3 
7, 7 
6, 14 
9, 5 

Payoffs  the first number in each cell is the payoff to the Row player, while the second number is the payoff to the column player

The bottom right cell has 9, 5

Row player has three strategies, R1, R2, and R3

Column player has three strategies, C1, C2, and C3

The payoff is denoted by p, i.e. the profit

Strictly dominant strategy – can player i maximize his payoff , regardless of the other player

p
_{i}(S _{i}, S _{i}) > p
_{i}(S _{i}’, S _{i})

Notation

S _{i} is the best strategy for player i

S _{i} is the rival’s strategy

S _{i}’ is a strategy for player i

Dominant strategy

Row player selects R2

Column player selects C2

Thus, the payoff is 15 for the row player and 12 for the column player
3. Prisoner’s Dilemma

Criminal 2 
Criminal 1

Rat 
Clam 
Rat

2, 2 
0.5, 5 
Clam

5, 0.5 
1, 1 

Payoff is years in prison

Cell 4: This would be best outcome for both criminals, because each spend a year in prison

Police: If one tells, he gets a half year in prison, while the other criminal gets 5 years

What happens if both criminals are in the mafia?

Criminal 2 
Criminal 1

Rat

Clam

Rat

death, death 
death, 5 
Clam

5, death 
1, 1 
4. Nash Equilibrium – a strategy profile, so every player’s strategy is the best response to strategies of the other players

Criminal 2 
Criminal 1 
Rat 
Clam 
Rat 
2, 2 
0.5, 5 
Clam 
5, 0.5 
0, 0 
5. Battle of the Sexes

Woman 
Man 
Football 
Ballet 
Football 
10, 1 
0, 0 
Ballet 
0, 0 
1, 10 

If the woman chooses football, what is the man’s best strategy?

The man chooses football

If the man chooses football, what is the woman’s best strategy?

The woman also chooses football

A Nash equilibrium is FootballFootball

If the woman chooses ballet, what is the man’s best strategy?

This game has two Nash equilibria and one equilibrium is not preferable to another
6. Zero Sum Game – one player gains, while the other player loses

Column Player 
Row Player 
A 
B 
A 
1, 1 
3, 3 
B 
0, 0 
2, 2 

If the column player selects A, what should the row player do?

The row player selects B

If the row player selects B, what should the column player do?

The column player selects B

Different cells, thus no Nash equilibrium

If the column player selects B, what should the row player do?

The row player selects A

If the row player selects A, what should the column player do?

The column player selects A

Different cells, thus no Nash equilibrium

This example has no pure Nash equilibrium
7. A more complicated game

C1 
C2 
C3 
R1 
3, 6 
2, 2 
1, 10 
R2 
5, 7 
6, 8 
9, 1 
R3 
5, 1 
8, 3 
10, 15 
8. Another example

C1 
C2 
C3 
C4 
R1 
3, 5 
3, 10 
1, 3 
10, 7 
R2 
0, 0 
5, 2 
5, 6 
4, 1 
R3 
5, 8 
20, 19 
4, 5 
3, 4 
R4 
10, 9 
4, 3 
2, 0 
9, 6 

The Nash equilibria are (R4, C1), (R3, C2), and (R2, C3)
9. Mixed Strategies

No Nash equilibrium exists

If two players play the game over and over again, a player can randomly select a strategy part of the time that maximizes its payoffs

Two players, McDonalds and Burger King

Burger King 
McDonald's 
High Price 
Low Price 
High Price 
40, 30 
30, 35 
Low Price 
35, 25 
32, 20 
payoff = 30 P _{m} + 25 (1  P _{m})

P _{m} is the probability that McDonald's chooses the high price, while 1  P _{m} is the probability McDonald's chooses the low price

If Burger King chooses the Low Price, its expected payoff is:
payoff = 35 P _{m} + 20 (1  P _{m})

Burger King wants to maximize its payoffs after it keeps playing this game

Set the two payoffs equal to each other, and solve for the probability, P _{m}
30 P _{m} + 25 (1  P _{m}) = 35 P _{m} + 20 (1  P _{m})
P _{m} = 0.5

Thus, McDonald's should charge a high price for half the time, and a low price for the other half

Similarly, McDonald's has the payoff, which is:
payoff = 40 P _{B} + 30 (1  P _{B}) = 35 P _{B} + 32 (1  P _{B})
P _{B} = 2 / 7

P _{B} is the probability that Burger King chooses the high price

Burger King should charge high prices 2/7 of the time, and low prices for 5/7 of the time
10. Sequential Games  one player makes a move, and then the other player

If Ben chooses "to stay out," then Jerry can be aggressive or maintain current price

If Jerry chooses to be aggressive, then Ben "stays out" of the market

If Jerry chooses to maintain current price, then Ben enters the market

If Ben chooses "to enter" market, then Jerry will maintain current price

We could put this into a matrix below

Jerry 
Ben 
Aggressive 
Maintain current price 
Stay out 
0, 1 
0, 1 
Enter 
0.5, 1 
1, 0 

What makes this example odd is it is missing a cell

Monopoly would never choose to fight entry

Nash equilibrium is green

Monopoly 
New Firm 
Fight entry 
Continue to max profit 
Stay out 

0, 80 
Enter 
0, 10 
30, 50 

1. Some students have difficulty understanding game theory. I present more examples. As you probably noticed, an interaction between two or more parties where each party has a choice can be structured as a game theory problem.
Example 1: The government provides a public good.
 Game Setup – Two people
 Each person has a maximum reservation price of $150 for the public good.
 The costs for government to supply public good is $150.
 If both people contribute, then the cost is $75 each.
 If one person contributes, then the cost is $150.
 The payoffs are the amount a person keeps after paying for public good, and are shown below.
(a) Identify the dominant strategies.
(b) Identify the Nash equilibriums if any.
Example 2: We have the battle between the sexes.
 A man and woman are dating
 One selects an activity and the other follows
 The payoffs are the utility for each person
 Joke – Man gets a utility of 5 when eating alone (Men love to eat!)
(a) Identify the dominant strategies.
(b) Identify the Nash equilibriums if any.
(c) Identify the Preferable Nash equilibrium.
Example 3: You have the payoff matrix below:

C1 
C2 
C3 
C4 
R1 
0, 7 
2, 5 
7, 0 
6, 6 
R2 
5, 2 
3, 3 
5, 2 
2, 2 
R3 
7, 0 
2, 5 
0, 7 
4, 4 
R4 
6, 6 
2, 2 
4, 4 
10, 3 
(a) Identify the dominant strategies.
(b) Identify the Nash equilibriums if any.
(c) Identify the Preferable Nash equilibrium.
Example 4: You have the payoff matrix below.

Left 
Middle 
Right 
Upper 
5, 4 
0, 1 
0, 6 
Middle 
4, 1 
1, 2 
1, 1 
Down 
5, 6 
0, 3 
4, 4 
(a) Identify the dominant strategies.
(b) Identify the Nash equilibriums if any.
(c) Identify the Preferable Nash equilibrium.
Example 5: We have a sequential game between the Cypress government and its citizens' bank deposits. The choices or strategies are:
 The Cypress government wants to impose a tax on bank deposits as a means to pay for the EU bailout package. Its choice is to tax or not tax bank deposits.
 The depositors have a choice. They can withdraw their deposits or keep funds at the bank.
 The payoffs are millions of euros, which are defined as (gov. tax revenue, depositors' wealth).
Identify the Nash equilibriums if any
Example 6: We have a sequential game between the Greek government and the European Union (EU). The choices or strategies are:
 The Greek government has a choice to withdraw from the Eurozone and reintroduce its currency.
 The EU has a choice to grant or not grant a loan to the Greek government.
 The payoffs are the change in GDP in millions of euros, which are defined as (Greece's GDP, EU's GDP).
Identify the Nash equilibriums if any
Example 7: We have a sequential game between a worker and a manager. Their choices are:
 The manager has a choice to fire or not fire his employee.
 The employee has a choice to arrive at work on time or clock in late.
 The payoffs are utilities, which are defined as (worker's utility, manager's utility).
Identify the Nash equilibriums if any
Example 8: We have a sequential game between a driver and the police. Their choices are:
 The driver has the choice to inform the police about his 4th Amendment rights, or consent to his car being searched.
 The police can pull the driver out of the car and beat him mercilessly or the police can respect the driver's rights.
 The payoffs are utilities, which are defined as (driver's utility, police's utility).
Identify the Nash equilibriums if any
2. Solutions
Example 1: The dominant strategy for both players is not to contribute for the public good. Consequently, the Nash equilibrium is Doesn't contributeDoen't contribute. As you guessed, people want benefits from their government but they do not want to pay for them.
Example 2: The players do not have any dominant strategies. The Nash equilibriums are DinnerDinner and MovieMovie. The MovieMovie is preferable because both parties experience greater utility.
Example 3: The players do not have any dominant strategies. The Nash equilibrium is R2C2, and no preferable Nash equilibrium exists. The game has only one Nash equilibrium.
Example 4: The Row player has a dominant strategy, Down, while the Column player has no dominant strategy. The Nash equilibriums are MiddleMiddle and DownLeft. The preferable Nash equilibrium is the DownLeft because it yields a greater payoff.
Example 5: The Nash equilibrium is the government taxes bank deposits and the depositors withdraw their bank funds.
Example 6: The Nash equilibrium is the Greek government remains in the Eurozone and the EU grants loans to Greece.
Example 7: This game has two Nash equilibriums: The first is the worker arrives on time and manager does not fire. The second is the worker arrives late and the manager fires the worker.
Example 8: I apologize for this problem, but I thought it was good gametheory example to illustrate the deteriation of people's rights in the United States. The first Nash equilbrium is the driver informs the police of his right, and the police pull the driver from the car and beat him. The second Nash equilibrium is the driver allows the police to search his car and the police respect his rights.
