Economics 650

1
Economics 650

• Game Theory

2
Probability
Probability a numerical measure of the
likelihood of one of the outcomes of an uncertain
event, which corresponds to the relative
frequency of that outcome if the event can be
repeated an unlimited number of times.
3
Example
When we throw a die, the probability of a number
greater than five is one sixth, while the
probability of a number greater than three is one
half. Thus, the more likely statement has the
larger probability.
4
Why?

• Why do we say that the probability of a number
  greater than three is one half?
• Three of the six possible outcomes give numbers
  greater than 3.
• If we throw the die many times, the relative
  proportion of throws giving a number greater than
  six will approximate 1/2.

5
Subjective Probability

• Dice are an example of objective probability.
• It seems that success in a research project to
  produce automobile engines using fuel cell
  technology would be more probable than success in
  a research project to produce automobile engines
  using nuclear fusion.
• This judgment is at least partly subjective.

6
Estimating Probabilities
The records show that about 20 of Christmases in
the Philadelphia area have been white
Christmases. So we can say with good confidence
that the probability of a white Christmas is
about 20.
7
Expected Value
Definition Expected value Suppose an uncertain
event can have several outcomes with numerical
values that may be different. The expected value
(also known as mathematical expectation) of the
event is the weighted average of the numerical
values, with the probabilities of the outcomes as
the weights.
8
Example

• You are to bet on the fall of a die.
• The bet pays 10 if the die comes up six, nothing
  otherwise.
• We have 1/6 10 5/6 0 for a total of 1.67.
• Thus, 1.67 is the expected value of the gamble.

9
Guys and Dolls
In the musical comedy Guys and Dolls, the leading
character, Sky Masterson, is a notorious gambler.
According to a description, he would bet on which
of two raindrops dripping down a windowpane would
reach the bottom first. This is a good example of
natural uncertainty.
10
Natural Uncertainty
Definition Natural Uncertainty uncertainty
about the outcome of a game that results from
some natural cause rather than the actions of the
human players.
11
Nature as a Player
When there is natural uncertainty, the convention
in game theory is to think of "nature" as a
player in the game. But "Chance" or "Nature"
doesn't care about the outcome and always plays
her strategies at random, with some given
probabilities.
12
Example

• Suppose that a company is considering the
  introduction of a new product.
• Unknown technological and market conditions may
  be "good" or "bad" for the innovation.
• This is "Nature's" play in the game.

13
Example, Continued

• Nature plays a 50-50 probability rule.
• The firm's strategies are go or no-go.

14
Value of Information 1

• What is the advantage of having more information?
  
• The advantage is that, with more information, we
  can make use of contingent strategies.

15
Value of Information 2
With the second strategy the e.v. payoff is
doubled from 5 to 10 -- the information is worth
5.
16
A Naval Example

• The British navy fought the Spanish at the Battle
  of Cape St. Vincent.
• The British commander decided to split the
  Spanish, sailing between two groups.
• If the wind were to increase, it would help the
  Spanish more than the British.

17
Naval Example in Normal Form
18
Dominant Strategy Equilibrium

• Each side maximizes its expected value by
  choosing an aggressive strategy.
• That is what the two sides did, except for one
  Spanish ship that left the line of battle and ran
  for a safe port to the east.

19
Expected Value Payoffs
20
Summary 1

• Probability is a way of expressing the relative
  likelihood of an uncertain statement in numerical
  terms.
• In game theory, uncertainty is customarily
  brought into the picture by making chance or
  nature one of the players in the game.

21
Summary 2

• If we know that the payoff in a decision or game
  will be one of several numbers, and we can assign
  probabilities to the numbers, we make compute the
  expected value of the pay off as the weighted
  average of the payoffs numbers, using the
  probabilities as weights.

22
It Can Be Rational to be Unpredictable
The players may deliberately introduce
uncertainty into the game, because it suits their
purposes. There are some games in which the best
choice of strategies is an unpredictable choice
of strategies. A good example is found in the
great American game of baseball.
23
Payoffs for Baseball
24
It is Rational to be Unpredictable

• If the pitcher always throws his best pitch, the
  batter will always swing early, and get a lot of
  hits.
• If the batter swings early on every pitch, then
  the pitcher can just throw a changeup on every
  pitch. The batter, too, needs to be
  unpredictable.
• Each of the two baseball players will choose
  unpredictably between their two strategies.

25
Mixed Strategy
Each will choose one strategy or the other with
some probability. This is called a mixed
strategy, since it mixes the two "pure
strategies" shown in the table.
26
Mixed and Pure Strategies
Definition Pure and Mixed Strategies In a game
in normal form, a player who chooses among the
list of normal form strategies according to given
probabilities, two or more of which are positive,
is said to choose a mixed strategy. A normal form
strategy chosen with certainty is called a pure
strategy.
27
But what probabilities will they choose?
From my point of view, the key idea is to prevent
the other player from exploiting my own
predictability. So I want to choose a probability
that will keep him guessing.
28
Let p be the probability that the pitcher throws
a fastball.
These are payoffs to the batter.
15p-53-8p p8/23
29
Intellectual Jiu-jitsu
It may seem odd that the pitcher is balancing
the expected value payoffs for the batter and
equalizing them. But the batter will shift to
pure strategies whenever p is the least bit
different from 8/27.When both the batter and the
hitter choose their best probabilities, each is
choosing the best response -- we have a Nash
equilibrium in mixed strategies.
30
Business Example A Blue-Light Special
Some sales seem to be unpredictable. For example,
there may be a sale without any notice when a
blue light is turned on. Why would a retailer
want her sales to be unpredictable? This might be
a mixed strategy.
31
A Simple Example
A two-person game The seller is one player and
the consumer is the other player. The seller's
strategies are to schedule a sale today or
tomorrow. The consumer's strategies are to visit
the store today or tomorrow.
32
Payoff Table
33
Expected Value Payoffs for Seller Strategies
8-3p 46p solving, p4/9
34
Equilibrium
The consumer will come today with probability 4/9
and tomorrow with probability (1-4/9) 5/9. We
conclude that this seller will schedule a sale
for today with a probability 1/3 and for tomorrow
with a probability 2/3.
35
Nashs Breakthough
In previous weeks, we have seen that some games
do not have Nash equilibria in pure strategies.
However, all two-person games have Nash
equilibria when we allow for mixed as well as
pure strategies. This was Nashs discovery, the
one for which he got the Nobel Memorial Prize.
36
New Example
The Gotham Bats, a major-league baseball team,
are threatening to move their franchise from
Gotham to Metropolis, if Gotham does not build
them a new stadium. If the Bats do move, the
city government will have political problems, and
they will be even worse if they have failed to
build the stadium. On the other hand, the stadium
is costly, and the best outcome for the
government is to keep the Bats and not build the
stadium. Thus we have the following 2x2 game.
37
New Example Game
Determine all Nash equilibria.
38
Mixed and Pure Strategies
39
Equilibria

• This game has two Nash equilibria in pure
  strategies.
• Whenever one player goes and the other waits, we
  have a Nash equilibrium.
• However, this game also has a mixed strategy
  equilibrium.
• Each plays wait with probability 2/3 and go with
  probability 1/3.

40
Inferior Equilibrium
Alphonse and Gaston can do no worse than 2 in a
pure-strategy equilibrium, but the expected value
payoff is 1 in the mixed strategy. Whats going
on here?
41
Unstable Equilibrium
Wait -- dark Go -- gray
42
Stable Equilibrium -- Baseball
Batter early -- dark late -- gray
43
Expected Value Payoffs in the Advertising Game
Advertising game Dont -- dark Do -- gray
44
Summary

• Uncertainty may come from the human players in
  the game.
• When strategies are chosen probalistically, we
  call it a mixed strategy.
• all two-person games have Nash equilibria, when
  mixed strategies are included.