Economics 650

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Economics 650

• Game Theory

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N-Person Games
Many of the "games" that are most important in
the real world involve considerably more than two
or three players -- for example, economic
competition, highway congestion, overexploitation
of the environment, and monetary exchange. BUT --
if we have ten players, there are 10! 3,628,800
relationships between them.
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Example The Queuing Game
Have you had this experience? Six people are
waiting at an airline boarding gate, awaiting
their chance to check in, and one of them stands
up and steps to the counter to be the first in
the queue. As a result the others feel that they,
too, must stand in the queue.
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Explanation
Yes, there is a game theory explanation -- and it
can also be applied to research and development,
innovation, and intellectual property.
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Payoffs 1

• There is a two-point effort penalty for standing
  in line, so that for those who stand in line, the
  net payoff to being served is two less that what
  is shown in the second column.
• Those who do not stand in line are chosen for
  service at random, after those who stand in line
  have been served.

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Payoffs 2
If no-one stands in line, then each person has an
equal chance of being served first, second, ...,
sixth, and an expected value payoff of
(1/6)20(1/6)17 (1/6)5 12.5. In such a
case the aggregate payoff is 75. But the net
payoff to the person first in line is 18gt12.5, so
someone will get up and stand in line.
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Payoffs 3
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Equilibrium
This game has a large family of Nash equilibria,
so we proceed by elimination. With 4 persons in
the queue, we have arrived at a Nash equilibrium
of the game. The total payoff is 67, the expected
value payoff is 6.5 for those who remain. Since
the fifth person in line gets a net payoff of 6,
no-one else will join the queue.
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Cui Bono?
Two people are better off -- the first two in
line -- with the first gaining an assured payoff
of 5.5 above the uncertain expected value payoff
she would have had in the absence of queuing and
the second gaining 2.5. But the rest are worse
off. The third person in line gets 12, losing
0.5, for example.
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Simplifying Assumptions for N-Person Games

• In the Queuing Game, all of the participants are
  assumed to be identical, to be "representative
  agents.
• The length of the line is a state variable. A
  state variable is a variable or vector that sums
  up the state of the game from the point of view
  of the representative agent.

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The Choice of Transportation Modes -- Car or Bus
The commuters are representative agents their
payoffs vary in the same way with the number of
cars on the road and the state variable is the
proportion of all commuters who drive cars rather
than riding the bus. The larger the proportion
who drive their cars, the slower the commute will
be, regardless which transport strategy a
particular commuter chooses.
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Payoffs -- Dominant Strategy
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Payoffs -- Perhaps More Realistic
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The Tragedy of the Commons
In general, "the tragedy of the commons" is that
all common property resources tend to be
overexploited and thus degraded, unless their
intensive use is restrained by legal,
traditional, or (perhaps) philanthropic
institutions. The classical instance is common
pastures, on which, according to the theory, each
farmer will increase her herds until the pasture
is overgrazed and all are impoverished.
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Hawk vs. Dove, Revisited
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The Many-Creature Case 1

• Suppose that birds are chosen from a large
  population and randomly matched to play the Hawk
  vs. Dove game.
• Birds do not choose to be Hawks or Doves but a
  certain proportion of the population are Hawks or
  Doves by genetic constitution.
• The chance of being matched with a Hawk is the
  same as the proportion of the population who are
  hawks.

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The Many-Creature Case 2
Expected Value Payoffs With Random Matching
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The Many-Creature Case 3

• Payoffs are the inclusive fitness of the two
  types.
• We see that the stable equilibrium of the
  population of hawks and doves corresponds to the
  mixed-strategy equilibrium of the Hawk vs. Dove
  game.
• This random matching model transforms the
  two-person game into an N-person game and is a
  common convention in applications to biology.

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Types
Assumption Types of Representative Agents In
some game analyses we may assume that there are a
small number of different types of representative
agents
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Tradition! Tradition!
Many traditional models in economics, both
classical and Keynesian, use the representative
agent approach -- and sometimes overuse it!
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Summary

• N-person games can be complex. Many applications
  in economics, political science and other fields
  rely on two complementary simplifying
  assumptions
• Representative agents and
• One or more state variables

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Alternatives to Nash Equilibrium

• Nash Equilibrium is a key concept for
  noncooperative games, but it is not quite the
  whole story.
• Other concepts of noncooperative solution to
  consider are
• Refined Nash Equilibria
• Correlated Strategies
• Rationalizable Strategies
• For this discussion we consider only games in
  normal form.

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Sibling Rivalry, Again

• Recall the Sibling Rivalry Game, which has two
  Nash equilibria math, lit and lit,math.
• Can we narrow the field by ruling out one or the
  other?

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Equilibria

• The grade point game between Iris and Julia has
  two Nash equilibria.
• The strategies (math, lit) give both girls
  perfect 4.0 averages, while the other Nash
  equilibrium, (lit, math), leaves Julia with only
  a 3.8.
• IEDS would eliminate the second -- but it is a
  Nash Equilibrium.
• Common sense might also eliminate the second
  Nash equilibrium for the failsafe 4.0, 4.0

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Refinement
Further assumptions might refine Nash
equilibrium to eliminate the questionable
equilibrium. Suppose Julia allows for a very
small probability that her opposite number will
make a mistake. We will see that the unreasonable
Nash equilibrium is eliminated. This assumption
is called the trembling hand assumption. This
narrows the possibilities to a single equilibrium
that seems to be more reasonable than the other.
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Trembling Hand 1

• Julia supposes that Iris might choose lit -- by
  mistake -- with a small probability p. Then her
  expected values for her two strategies are
• For lit 3.8(1-p)3.7p3.8-0.1p
• For math 3.8(1-p)4.0p3.80.2p
• For any positive p, math has the higher expected
  value.
• Thus, if Julia thinks Iris might have a
  trembling hand, lit,math can be ruled out as
  a rational solution.

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Trembling Hand 2

• Iris has equally good reason for concern if Julia
  might have a trembling hand.
• In general if a,b is a Nash Equilibrium, and
  for sufficiently small probabilities of mistakes
  (trembles) a and b both have larger expected
  values than other strategies, then a,b is a
  trembling hand NE.
• In the Sibling Rivalry game, math, lit is the
  only trembling hand NE.

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Interim Summary 1

• In games with more than one Nash equilibrium, we
  may want to eliminate some equilibria from our
  consideration by means of refinement.
• A refinement is an additional assumption about
  rationality that serves to rule out some
  equilibria as irrational.
• An example is the trembling hand test.
• If an equilibrium is not fail-safe against a
  small probability of errors by the other player,
  it fails the trembling hand test.
• An example is the unreasonable equilibrium in
  the Sibling Rivalry game.

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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

• In the Movie, Nash suggests that they each pursue
  a brunette, so that all can expect companionship.
  
• The movie is fiction, and the game theory and
  economics in it are not reliable.
• There are two Nash equilibria, at each of which
  just one male hits on the blonde.
• But can they achieve a Nash equilibrium in this
  case? There is a risk that each assumes the other
  will settle for a brunette -- resulting in 0,
  0, a coordination impasse.

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Coalition
But here is a symmetrical solution that avoids
the risk of a coordination impasse. The males
form a grand coalition to coordinate their
strategies first, they assign different roles
among themselves. One of them will be "it" and
the others will all be "non-it." The roles are
assigned by some method that gives each male an
equal probability of being "it. Second, the one
who is selected as "it" has a free pass to
approach the blonde. All of the others are
expected to court the brunette of their
respective choice.
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Nash Equilibrium
Can anyone improve his payoff by a unilateral
switch of strategies? First, consider "it." His
payoff following the agreed strategy is 2.
Otherwise it is 1. He has no reason to deviate.
What about a "non-it?" If he were to deviate by
pursuing the blonde, he can be certain of facing
competition from "it." In that case, his payoff
would be zero. However, by playing the strategy
assigned to him at the first stage, he can assure
himself of a payoff of 1. Thus the agreement is
self-enforcing.
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Correlated Strategies

• Notice what they have done
• They have randomized their strategies -- but
  jointly, assigning positive probabilities to Nash
  equilibria and zeros to coordination impasses.
• Thus their strategies are correlated -- by
  contrast with mixed strategy equilibria which are
  independent.
• This is called a correlated strategy solution.

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The Nathan Detroit Dilemma
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Equilibria?

• There are no Nash equilibria in pure strategies.
• The mixed strategy Nash equilibrium is very
  unfavorable to the coalition of gamblers
• Nathans job is to correlate their strategies

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Mall Location Game 1

• Yall-Com Computers and Lotsal Lectronics are
  considering locations in the Mall or at
  stand-alone locations.
• The Mall would like to get them both, but that is
  not an NE.

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Mall Location Game 1

• The only way that that Groundswell Mall can
  influence their decision is by sending one or
  more of the store chains a special invitation.
• The special invitation is cheap talk.
• Groundswell will randomize their invitation
  strategy, sending special invitations to both
  companies with probability ½ , sending a special
  invitation only to Lotsa Lectronics with
  probability and sending a special invitation
  only to Yall-Com Computers with probability . The
  probability that they send no special invitations
  at all is zero.

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Mall Location Game 2

• Groundswell announces the probabilities it will
  use, and requests each company to keep it
  confidential if they do get a special
  invitation.
• Each company considers making its decision
  according to Rule A If I receive a special
  invitation, choose the Mall, and otherwise
  choose the stand-alone location.
• This proves to be a Nash equilibrium of the
  larger game expanded by the messages!

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Mall Location Game 3

• The expected value payoffs for the two companies
  are both
• better than they can do with a flipped coin
  correlated strategy solution.
• Groundswell Mall also does better, since they
  will get at least one store and have a
  fifty-percent chance for both.

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Interim Summary 2

• Correlated equilibria provide an alternative for
  coalitions in coordination games.
• In effect, the coalition chooses a joint mixed
  strategy.
• This may assign a zero probability to bad
  strategy combinations, and (with different
  signals to the two players) induce them sometimes
  to choose non-equilibrium good strategy pairs.

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Rationalizable Strategies
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What Happened, Charlie Brown?

• In this 1962 strip (rerun Oct. 4, 2009) Charlie
  Brown did his best to outthink Lucy, but made a
  mistake.
• Youd think he would choose his best response,
  having played this game many times.
• But -- if he were playing for the first time --
  it would be understandable to make an error of
  this kind.
• Charlie was attempting to find a rationalizable
  strategy.

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A Surfy City Real Estate Game
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Rationalizable Strategies in this Game 1

• The game has no Nash equilibria in pure
  strategies. It will only be played once, so
  (unlike Charlie Brown) Laura and Mark will have
  no opportunity to correct their errors.
• Laura reasons, He thinks I think he will choose
  office, and that I will therefore open a parking
  lot. Accordingly, he will build a restaurant.
  Thus I will maximize my payoff by building a
  cinema.
• Mark reasons, She thinks I think she will build
  a cinema, so I will operate an office complex.
  But that will lead her to open a parking lot, and
  I will respond by opening a restaurant.

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Rationalizable Strategies in this Game 2

• This reasoning leads them to build a cinema and a
  restaurant, respectively. In this case, each
  person has chosen a rationalizable strategy.
• Each based her or his decision on a conjecture, a
  guess, as to how the other person interprets the
  guessers decision process.
• parking lot and office complex are also
  rationalizable strategies, since they are best
  responses to strategies that the players might
  conjecture that the other would choose. But the
  other strategies are not rationalizable.

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Rationalizable Strategies in General 1

• If a particular strategy can be justified as
  these two strategies were that is, if 1) the
  first player starts out with a conjecture about
  what the second player believes that the first
  player believes about the second players
  choices, and 2) this conjecture cannot be known
  to be false until the game is played, and 3) the
  first player predicts that the second player will
  play a strategy consistent with this belief, and
  4) the first player plays her or his best
  response to that strategy, then the strategy
  played by the first player is a rationalizable
  strategy. A rationalizable strategy solution is
  one in which both players play rationalizable
  strategies.

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Rationalizable Strategies in General 2

• There is a rather simple and intuitive method for
  the determination of rationalizable strategies.
• First, let us define an irrelevant strategy as
  a strategy that is never a best response, no
  matter what strategies the other player chooses.
• Now we will iteratively eliminate all irrelevant
  strategies. Those that remain are rationalizable.

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A Television Programming Game
Local news is irrelevant for both players.
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Elimination, Step 1
Network news is now irrelevant for WACK.
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Elimination, Step 2
Dr. Nice is now irrelevant for KRUM.
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Elimination, Step 3
We find that this game has a NE in pure
strategies and those are the only rationalizable
strategies. (NE strategies are always
rationalizable.)
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Interim Summary 3

• Rationalizable strategies make sense if the
  game is played only once, so errors cannot be
  corrected.
• In such a solution, each person makes a
  conjecture about how the other expects him to
  choose, and what the others best response to
  that expectation is, and chooses her own best
  response to that best response.

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Overall Summary

• Nash equilibrium may not be the only, or the
  best, solution in a noncooperative game.
• When there are plural NEs, we may want to
  narrow the field by refinement.
• On the other hand, we may broaden the field by
  signals that create a larger game with a joint
  mixed strategy.
• Alternatively, we may be able to do no better
  than a rationalizable strategy, that is, one
  that is based on a guess about what the other
  player guesses I will do.