Consider the game of matching pennies. One player is "match-maker" and the other is "variety-maker."
Each player takes a penny and secretly places it heads up or tails up. Then the players reveal their
pennies. If they are the same, "match-maker" gets both. If they are different, "variety-maker" gets
both.
Fill in the payoff matrix for this game (in the cells, the comma separates the payoffs for the row player
and the column player.
Variety- Maker
Heads
Tails
Heads
,
,
Tails
,
,
Match-Maker
Why might the term "zero-sum" apply to this game?
Consider the following game matrix.
Player B
C
D
C
0,0
-2, 1
D
1, -2
-10,-10
Player A
You are player A. Suppose you think the other player will choose C, what should you do?
Suppose you think the other player will choose D, what should you do?
You have your first job after college. Things have been going well for you on the job for the last year and
a half. You think about asking for a raise. Should you? What can you lose? Well, face, for one. It would
certainly be worse, you decide, to have to come to work after having been turned down than it would
be to have never asked at all.
1. Construct the matrix for a "Matching Nickels with Pennies from Heaven" game, played as
follows. Players play "Matching Nickels (just like matching pennies but with nickels) except no
matter what the outcome, each player receives a penny from heaven. Thus, a player ends up
with either 5+1 = 6¢ or -5+1=-4¢. Make "match-maker" the row player and "variety-seeker" the
column player.
a. Show the payoff matrix with both players' numbers.
Variety- Maker
Heads
Tails
Heads
,
,
Tails
,
,
Match-Maker
b. Add up the two numbers in each cell. What do you notice? What is the sum?
c. This is not a zero-sum game but it is a constant sum game.
d. Constant-sum games are "strategically equivalent" to zero-sum games. The pennies
make everyone just a little bit wealthier, but wealth is not a part of the game
description and so does not change the game.
e. Construct another constant-sum game matrix using whatever numbers come to mind.
Player B
Move 1
Move 2
Move 1
,
,
Move 2
,
,
Player A
f.
Construct the equivalent zero-sum game.
Player B
Move 1
Move 2
Move 1
,
,
Move 2
,
,
Player A
Is the following game zero sum, constant sum, or variable sum?
Player B
C
D
C
0,0
-2, 1
D
1, -2
-10,-10
Player A
RULE 1. If you have a dominant strategy, use it.
There can be an equilibrium even if one player does not have a dominant strategy. Consider this
matrix. Only one number is in each cell – it'sthe row player's payoffs – because the column players
are the reverse in this zero sum game.
Player B
-2
1
-3
4
Player A
Does A have a dominant strategy?
Does B have a dominant strategy?
Given this what will A do? What will be the results?
Rule 2. Assume other player is rational and respond with best move to what she will do.
Equilibrium: Outcome such that if anyone had acted differently, she would have done worse. OR,
everyone has made her best choice. No regret.
Player B
Player A
5
0
-3
3
2
4
-2
1
6
Parking Problems
Tragedy of the commons. Mix of families with 0, 1, 2, 3 cars. Mix of space in front of one's house.
Some houses with driveways and garages.
Suppose parking in the driveway is a little inconvenient and that the space in front of the average
house has room for between 1 and 2 cars but not enough for two. Thus, parking both cars on the
street takes up a bit of the house next store's space. Parking close means you have no
inconvenience, parking far means you have 5 units of inconvenience.
How many cars do we park right in front?
Player B
One
Both
One
5, 5
5, 0
Both
0, 5
conflict
Player A
What about roommates?
Can we define a generic roommate game? Or is it a series of games?
Simple work/shirk games.
2.8 Tragedy Of The Commons
Consider a village with N farmers, that has limited grassland. Each of the N farmers
have the option to keep a sheep. Let the utility of milk and wool from the sheep be 1. Let
the damage to the environment from one sheep grazing over the grassland be denoted by –
5.
Let Xi be a variable that takes values 0 or 1 and it denotes that whether the ith farmer
keeps sheep or not.
Xi = 1, i
th
farmer keeps sheep
= 0, i
th
farmer does not keep sheep
So, the utility of the i
th
farmer, Ui(Xi), is given by :Ui(Xi) = Xi – { [ 5 * (X1 + X2 + X3 + ..... Xn) ] / N }
since, the total environmental damage is shared by all farmers.
In this game,
Xi = 1, is a Nash Equilibrium for all i, if N >= 5
and this Nash Equilibrium is unique if N > 5. This can be easily shown because keeping
a sheep would add more utility to a farmer from milk/wool than subtract utility from him
due to environmental damage. Thus, everyone will end up keeping a sheep and the utility
for every farmer will be –4.
In real life this leads to excessive environmental damage. In this case, game theory
comes to rescue to by justifying a pollution tax of keeping the sheep, of an amount equal
to the damage done to the environment. So, when this pollution tax is added , the utility of
the i
th
farmer becomes
Ui(X1,X2,...Xn) = Xi - 5Xi – { [5 * (X1 + X2 + X3 + ..... Xn) ] / N }
US / USSR
Health
Defense
Health
10 , 10
-10 , 15
Defense
15 , -10
1,1
Page 6
Now, in this case, the Nash Equilibrium is Xi = 0, for all i. This is because any farmer
will increase his utility by giving away his sheep. Thus everyone will give away his sheep
and each will have utility of 0.
The above example of tragedy of the commons can be used to explain large number of
real life situations. Consider, the scenario of industries polluting the environment. Here,
the industries can be considered as sheep and the environment can be considered as grassland and similar analysis of tragedy of commons holds true. In another scenario of illegal
construction of houses causing infrastructure problems, the houses denote the sheep and
the infrastructure denotes the grassland in the tragedy of the commons.
Thus, game theory can be used to design laws and mechanisms to get socially desirable
outcomes