Economics 650

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Economics 650

• Game Theory

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A Quiz Game
If one contestant buzzes, she gets to answer the
question. A correct answer scores one point. The
contestant who does not buzz is penalized one
point. If neither contestant buzzes, the question
is passed, with no score for anyone. If both
contestants buzz, the question is passed with no
score for anyone.The questions are so easy that
both players are confident that they can answer
every question.
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The Buzzer Game
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About This Game

• It is easy to see that this game has a dominant
  strategy equilibrium (buzz, buzz).
• The total payoffs for the two players always add
  up to zero. That means this is a zero-sum game.

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Maximize the Minimum Payoff
Pamelas Payoffs
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Definitions
Definition Zero-Sum Game A game in which the
payoffs for the players always adds up to zero is
called a zero-sum game.
Definition Maximin strategy If we determine
the least possible payoff for each strategy, and
choose the strategy for which this minimum payoff
is largest, we have the maximin strategy.
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A Further Definition
Definition Constant-sum and nonconstant-sum game
If the payoffs to all players add up to the
same constant, regardless which strategies they
choose, then we have a constant-sum game. The
constant may be zero or any other number, so
zero-sum games are a class of constant-sum games.
If the payoff does not add up to a constant, but
varies depending on which strategies are chosen,
then we have a non-constant sum game.
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Rule
For a constant-sum game, the maximin solution is
the Nash equilibrium and the unique
game-theoretic solution.
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The Spring Water Game
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Dominated Strategies 1

• Recall, if a strategy is the best response for
  one player, regardless of the strategy the other
  player may choose, this strategy is said to be
  dominant.
• In that case another strategy is dominated.

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Dominated Strategies 2
Definition Dominated Strategy -- Whenever one
strategy yields are higher payoff than a second
strategy, regardless which strategies the other
players choose, the second strategy is dominated
by the first, and is said to be a dominated
strategy.
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Retail Location Again

• This game has a Nash equilibrium where each store
  chooses Center City.
• Even though this game has no dominant strategies,
  it does have some dominated strategies. (That's
  dominated, not dominant.)

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The Location Game with Dominated Strategies Shaded
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A Reduced Location Game with its Dominated
Strategies Shaded
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A Further Reduced Location Game with its
Dominated Strategies Shaded
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The Location Game with all Dominated Strategies
Iterativly Eliminated
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IEDS
Method Iterated elimination of dominated
strategies -- If a game has a dominated
strategy, the game created by the elimination of
that dominated strategy has the same Nash
equilibria as the original game. This elimination
can be done step by step until there are no more
dominated strategies, and the resulting game has
the same Nash equilibria as the original game.
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It Doesnt Always Find the Nash Equilibrium
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But IEDS Limits the Field
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Maximin Solution
In this game, Coolwater's minimum payoff at a
price of 1 is zero, and at a price of 2 it is
-5000, so the 1 price maximizes the minimum
payoff. The same reasoning applies to Springy
Springs, so both will choose the 1 price.
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In this game, Nash and Maximin Disagree
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But in this one, Maximin leads to a Very Bad
Outcome
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Classical Cases

• Although within the Nash family, these
  examples have special properties and have been
  widely studied.
• They are part of the language of game theory. We
  will see them again and again.

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The Battle of the Sexes
Marlene and Guillermo would like to go out
Saturday night. Guillermo would enjoy a baseball
game, while Marlene would prefer a show. Mostly,
they want to go together. They can't contact one
another because the telephone company is on
strike, and the e-mail system has crashed. Each
one can choose between two strategies go to the
game or go to the show.
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Payoff Table
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Equilibria
This game has two Nash Equilibria (game, game)
and (show, show). Once again, there is the
problem of determining which equilibrium is more
likely to occur. We don't have a Schelling point
to rely on. Because of its enigmatic nature, the
battle of the sexes game has played an ongoing
part in game theoretic research, and we will see
it again
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The Chicken Game
The chicken game is based on some hot rod movies
from the 1950's. The players are two hot rodders.
The game is one in which they drive their cars
directly at one another, risking a head-on
collision.
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Chicken Rules
If one of them turns away at the last minute,
then the one who turns away is the loser--he is
the "chicken." However, if neither of them turns
away, they both stand to lose a great deal more,
since they will be injured or killed in a
collision. For the third possibility, if both of
them turn away, neither gains or loses anything.
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Payoff Table
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Enigma
This game has two Nash equilibria, one each where
one hot rodder turns away and the other one goes
forward. But yet again, with two Nash equilibria,
and no signal or clue to define a Schelling focal
point, there is no way to say which of the two
equilibria is more likely.
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Hawk vs. Dove
Another example comes to us from biologists who
study animal behavior and its evolutionary basis.
It is called Hawk vs Dove. The idea behind this
game is that some animals can be quite aggressive
in conflicts over resources or toward prey, while
others make only a show of aggression, and then
run away.
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A Two-by-Two Game
In population biology, the assumption is that
creatures meet one another more less at random,
and dispute over some resource, using the
strategies of aggression or running away. The
Hawk vs. Dove game is played out at each meeting.
Payoffs are in terms of inclusive fitness.
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Payoffs
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Contrast

• The Hawk vs Dove game, like Chicken, has two Nash
  Equilibria, which implies some uncertainty.
• These games have applications in biology and
  international relations

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Stag Hunt Game

• Two Nash equilibria -- one payoff dominant, the
  other risk dominant.
• No dominant strategies
• Therefore, no dominant strategy equilibrium.

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Interim Summary

• Zero-sum games may have relatively simple
  solutions but can be misleading when applied to
  the real world.
• There are several families of two-by-two games,
  depending on the order (best, second best,
  worst) of the payoffs.
• These games illustrate some fundamental issues in
  game theory.

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Industrial Pricing
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Pricing Strategies
Game theory models of pricing strategy go back
before game theory -- in the 1840s,
mathematician Augustin Cournot proposed a model
of industry pricing that is now recognized as a
Nash Equilibrium model. But there was a century
of controversy over it
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So Far,

• In all of our games, each player has chosen from
  a finite and usually quite small number of
  strategies.
• We can learn something by modeling pricing games
  in that way, as we will see.

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More Realistically,

• In the real world, prices and output can vary
  over infinitely many possible levels.
• We can allow for this by using the concept of a
  reaction function.
• The reaction function generalizes the
  best-response table --
• The best-response output is a mathematical
  function of the output chosen by the other
  seller.
• Nash equilibrium is the intersection of the two
  curves.

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One of Cournots Ideas
Definition Demand curve or function -- The
relationship between the price of a good and the
quantity that can be sold at each respective
price is a demand relationship. It can be shown
in a diagram as the demand curve, or
mathematically as the demand function.
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Another of Cournots Ideas
Cournot assumed that each firm would decide how
much product to put on the market, and the price
would depend on the total. Thus, each firm has to
make a guess a "conjecture" as to what the
other firm will sell.
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Example

• We have two competing companies
• MicroSplat
• Pear Corp.
• Pear conjectures that MicroSplat will offer Q1
  for sale.

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In Graphical Terms
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Meanwhile, in Redmond
Of course, MicroSplat will also have to try to
conjecture about how much Pear will produce, and
determine their own demand curve and
profit-maximizing output in a similar way.
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Consistent Conjectures
Definition Consistent conjectures When two or
more decision makers each base their decisions on
conjectures about the decisions of others, and
the conjectures lead each decision maker to make
the decisions the others had conjectured that she
would so that everyone turns out to be right
we have a case of consistent conjectures.
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Maximizing Profits
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Each Seller Assumes

• ? The other seller will choose the "best-response
  strategy" (since they are both rational and have
  common knowledge of their rationality) so
• ? the other seller's output is given.
• ? Thus, each seller assumes he has the rest of
  the market to himself.

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Plan
The Cournot model is one of several traditional
models of duopoly prices that can be interpreted
in Nash Equilibrium terms. I will use a single
numerical example -- with the same industry
demand curve -- to compare and contrast them. The
demand relation can be written in algebraic terms
either as Q9500-5p or as p1900-0.2Q, where Q is
the total output of both firms and p is the price
they both receive.
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The Picture
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Reaction Functions
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Bertrand

• Bertrand (1883) asked why the sellers would focus
  on the outputs rather than compete in terms of
  prices.
• In game theory terms, Bertrand is suggesting that
  the prices, and not the outputs, would be the
  strategies.
• The key point is that if one seller cuts his
  price below the other, the seller with the lower
  price gets the whole market.
• If they charge the same price, the simple guess
  is that they split the market.

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Reaction Function

• In general this idea, the reaction function, can
  be applied whenever the strategies must be chosen
  from an infinite set, that is, a continuum
  (numerical interval).
• It has also been applied in monetary theory --
  what is the reaction function of the Fed to the
  rate of inflation, for example?

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Bertrand Example

• We will limit MicroSplat and Pear Corp. each to
  just three strategies Prices of 400, 700, or
  1000.
• Recall that 1000 is the monopoly price total
  profits will be greatest if they both charge that
  price. In our example, this leads to payoffs as
  shown in the table --

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Payoff Table
The lowest price is a dominant strategy.
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Cournot Model for the Same Game 1

• Once again, we will simplify by assuming only a
  limited number of possible strategies outputs of
  2000, 3000, 4000.
• For example, if MicroSplat chooses to produce and
  sell 2000, and Pear chooses to produce and sell
  3000, the industry total is 5000, and we see that
  the price in the industry will be 900.
• The next Table shows the payoffs, in millions,
  that will result from every pair of output
  strategies the two firms may choose.

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Cournot Model for the Same Game 2
Nash equilibrium is at intermediate output and
price.
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We see

• The assumption the prices are strategies leads to
  zero or minimal profits.
• The assumption that output is the strategy leads
  to positive profits, although below the monopoly
  level.
• Can we justify either assumption?

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Edgeworth

• Edgeworth (1897) pointed out two complications.
• First, the sellers may have limited production
  capacity.
• Second, Edgeworth argued that a supply and demand
  price would not be stable either.
• Edgeworth concluded that, if there are no
  capacity limits, there will be no stable price

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Edgeworth's Reasoning
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Nash Equilibrium
We can eliminate every strategy except the lowest
prices allowed, 6 and 6. In a game of this kind,
as long as there is any margin of price over
marginal cost whatever, the best response is
always to cut price below the other competitor.
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Product Differentiation

• When different firms sell products or services
  that are not perfect substitutes, and make the
  distinction among the products a basis of
  promotion or an aspect of market strategy, we
  refer to this as product differentiation.
• With product differentiation, each firm has a
  distinct demand function for its own product.
• However, they are likely to be interdependent, in
  that the cross-price elasticity is likely to be
  high.

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How Firm 1s Demand Varies with Firm 2s Price
(assumed)
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Firm 1s Demand

• Black if Firm 2 prices at 10
• Red if firm 2 prices at 20
• This will change the profit-maximizing price or
  output for firm 2.

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Reaction Function, Again

• Therefore, Firm 1s best price-and-output
  strategy will depend on the price-and-output
  strategy chosen by Firm 2.
• We can, again, represent this by a reaction
  function -- with either price or quantity.

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Price Reaction Functions
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Nash Equilibrium

• Once again, the Nash equilibrium is at the
  intersection of the reaction functions.
• If the two firms were merged, they would price
  both products at the green star.
• The cheaper production is a result of
  (monopolistic) competition.

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Calculus?

• In general, we will need to use calculus to get
  best responses, reaction functions, and Nash
  equilibrium.
• Differentiate profits by the strategy variable,
  price or quantity offered
• Set derivative equal to zero and solve
• Integrate to obtain reaction functions
• Solve simultaneous equations for the Nash
  equilibrium.

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Homeworks for Next Week

• Chapter 5 1, 2
• Chapter 6 2, 3

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Summary

• In a duopoly (or other oligopoly) determination
  of prices and outputs for maximum profits is an
  interactive decision.
• We can generalize the best response concept to a
  reaction function, applicable in this case.
• Remarkably, things are actually a little more
  predictable when products are differentiated.
• Price competition will still play a role, though
  perhaps a somewhat reduced one.