Economics 650

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Economics 650

• Game Theory

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Dominant Strategies in Games
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The Dumping Game
Mr. Jones and Mr. Smith own weekend homes on side
by side plots of land in a remote area that has
no routine garbage collection. They can contract
to have their garbage picked up by a trucking
firm, but it would cost either property owner
500 per year to arrange for the garbage pickup.
Alternatively, Mr. Jones can sneak across the
property line and dump his garbage on Mr. Smith's
property, and similarly, Mr. Smith can dump his
garbage on Mr. Jones's property.
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Two Vacation Plots
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The Game
The two landowners will make their decisions
simultaneously, so we will model it in normal
form. Each landowner chooses between two
strategies pay for a garbage pickup or dump the
garbage on the neighbor's property. But what are
the payoffs in this game? The benefits the two
landowners get from their weekend property are
subjective benefits -- but
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The Payoffs
We can estimate that the money value of a
person's subjective enjoyment of his property
it is the smallest amount of rent that he would
accept in return for giving up his own occupation
of the property. We will say that each person
values his experience at his weekend home at
5,000 per year if there is no dumping, but at
4,000 per year if there is.
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Payoff Table
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Rationality
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Rationality for Mr. Jones
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Dump is aDominant Strategy.
Dominant Strategy -- Whenever one strategy yields
a higher payoff than a second strategy,
regardless which strategies the other players
choose, the first strategy dominates the second.
If one strategy dominates all other strategies
(for a particular player in the game) it is said
to be a dominant strategy (for that player.)
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Dominant Strategy Equilibrium
Dominant Strategy Equilibrium -- If, in a game,
each player has a dominant strategy, and each
player plays the dominant strategy, then that
combination of (dominant) strategies and the
corresponding payoffs are said to constitute the
dominant strategy equilibrium for that game.
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Noncooperative Equilibrium
The dominant strategy equilibrium is a
noncooperative equilibrium. In this case,
though, Mr. Jones and Mr. Smith would probably
prefer a cooperative solution.
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Cooperative Equilibrium
Cooperative solution -- the cooperative solution
of a game is the list of strategies and payoffs
that the participants would choose if they could
commit themselves to an enforceable coordinated
choice of strategies.
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For Example,
Suppose, to continue the example, that Mr. Smith
and Mr. Jones come together and negotiate a
contract. The contract states that each of them
will hire a truck, and that there will be no
dumping. They could also specify that neither
would sell without a condition that the buyer
would continue the contract. In fact, contracts
of this kind among homeowners are quite common.
They are called "covenants."
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Social Dilemma 1.
The Dumping Game, the Prisoners Dilemma, and
Advertising Games are examples of Social
Dilemmas.
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Social Dilemma 2.
Social dilemma A social dilemma is a game that
has a dominant strategy equilibrium and the
dominant strategy solution is different from the
cooperative solution to the game.
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Another Dominant StrategyEquilibrium
Omnisoft Corp. and Microquip, Ltd. are
considering a collaborative project of research
and product development.
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Collaborative Product Development
Each company has two strategies to choose
between to commit plenty of resources to the
project, or to hold back. Neither partner can
enforce the commitment of effort and resources by
the other. In this case, however, we are assuming
that the project has "spinoff" technologies, that
is, technologies, and that will make a difference.
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Payoff Table
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No Dilemma!
The equilibrium at (commit, commit) is not only
the dominant strategy equilibrium in this game.
It is also the "cooperative solution," that is,
the outcome the players would choose if they
could make an enforceable agreement to choose any
pair of strategies at all.
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A Political Game
Strategic choices have to be made not only in
business, waste disposal, recreational games, and
war--but also in routine in peace-time politics.
Let's consider a political example. The
motivation is "Stimmungsmaximieren "--maximizing
the vote.
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Three Strategies
We shall assume that the voters are distributed
symmetrically, with 30 favoring the political
right, 30 favoring the political left, and 40
preferring to vote for a candidate with a
middle-of-the-road position.
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Payoff Table
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Best Response
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No Dilemma, But
This is not a social dilemma. Since the total
vote is the same, 100, regardless of which
strategies the two candidates choose, there is no
conflict between the dominant strategy
equilibrium and the cooperative solution in this
game. But that is looking at it from the point of
view of the candidates, and not from the point of
view of the voters.
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A Three-Strategy Dumping Game
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Best Responses
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Another Three-Strategy Game
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Best Responses
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No Dominant Strategies
This game does not have a dominant strategy
equilibrium because Dr. Boingboing's best
response strategy depends on the strategy Dr.
Heffalump chooses, and vice versa.
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Conclusion on Dominant Strategies

• A dominant strategy is the best response,
  whichever strategy the other chooses.
• If both players have dominant strategies, we have
  a dominant strategy equilibrium.
• If such an equilibrium exists, it is a powerful
  and persuasive concept of noncooperative
  solution.
• However, it may not always exist.

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Nash Equilibrium in Games
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Another Three-Strategy Game
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Best Responses
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No Dominant Strategies
This game does not have a dominant strategy
equilibrium because Dr. Boingboing's best
response strategy depends on the strategy Dr.
Heffalump chooses, and vice versa.
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Nash Equilibrium
Nash Equilibrium For any game in normal form if
there is a list of strategies, with one strategy
per player, such that each strategy on the list
is the best response to the other strategies on
the list, that list of strategies is a Nash
equilibrium. Note that (800, 800) is the Nash
equilibrium of the textbook writing game.
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Nash and Dominant Strategy 1

• Both reflect rational choice of strategies
• That is, best response on both sides
• Both are noncooperative
• But Nash is more general.

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Nash and Dominant Strategy 2
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Another ExampleBusiness Location

• choose one of four location strategies
• Uptown
• Center City
• East Side
• West Side

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Payoffs
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Equilibrium
The Nash equilibrium is found when both stores
locate in Center City.
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Finding Equilibria
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In-Class Exercise
Felix and Oscarina share their home with two
cats. Felix, who has a sharp sense of smell,
would like for the cat boxes to be cleaned twice
a week. Oscarina, whose sense of smell is less
acute, would be satisfied if they were cleaned
once a week. Each would prefer not to be the one
to clean the cat boxes. Their payoffs are shown
on the next slide
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Payoffs for the Catbox Game
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The Question
What is the Nash equilibrium for the Catbox game?
What is the cooperative solution? Who will empty
the cat boxes? Is this game a social dilemma? Why
or why not? Oscarina wants to get another cat,
but Felix says he will go along with the new pet
only if there is some agreement about cleaning
the cat boxes. Explain Felix' thinking.
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Nash -- both underlined, cooperative -- color
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Some Answers

• What is the Nash equilibrium for the Catbox game?
  
• Felix cleans twice
• What is the cooperative solution?
• Each cleans once
• Who will empty the cat boxes?
• Felix

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More Answers

• Is this game a social dilemma? Why or why not?
• No, because it does not have a dominant strategy
  equilibrium. Nevertheless the equilibrium is
  inefficient.
• Oscarina wants to get another cat, but Felix says
  he will go along with the new pet only if there
  is some agreement about cleaning the cat boxes.
  Explain Felix' thinking.
• Felix figures he will be cleaning three times.

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Choosing Radio Formats
(country, rock) and (rock, country) are both Nash
equilibria
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How can we choose between two Nash equilibria?
Schelling Point If a game has two or more Nash
equilibria, and some clue can lead the
participants to believe that one equilibrium is
more likely to be realized than the other, the
more likely equilibrium is called a Schelling
point.
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The Heave-Ho Game
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Paper, Stone and Scissors
This game has no Nash equilibrium in pure
strategies.
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Conclusion on Nash Equilibrium

• Nash equilibrium is a very general "rational
  solution" to a game, but it has some shortcomings
  from that point of view.
• The Nash equilibrium may not be unique.
• Not all games have Nash equilibria (in pure
  strategies).
• Nevertheless, it has wide applications.

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Homework Suggestions

• Ch. 3 2, 4, 7
• Ch. 4 2, 6

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The Blonde Problem
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A Puzzle from the Movie
In the cinema version of A Beautiful Mind, John
Nash reaches the insight that leads to his
Nobel-Prize-winning breakthrough by solving what
we might call The Blonde Problem. (Caution
Nash told my co-author Yvan Kelley that the
movie is fiction and the game theory and
economics is unreliable.)
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The Blonde Problem 1

• There are 2 or more lusty males.
• There are several possibly interested females.
  There are at least one more females than males.
• Just one female is blonde.
• Every male prefers a blonde to a brunette, and a
  brunette to no female companion at all.

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The Blonde Problem 2
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The Blonde Problem 3
What is the solution to The Blonde Problem? Dont
assume the movie got it right! Taint
necessarily so! What is the Nash Equilibrium
for this game? Hint could there be more than one?