This document discusses coordination games and continuous strategy spaces in game theory. It provides examples of coordination games with discrete and continuous pure strategies. For the continuous strategy case, it shows how to find the Nash equilibrium by deriving each player's best response function and setting them equal to solve for the equilibrium. The document discusses how multiple equilibria can arise and different approaches for selecting among them, such as focusing on the equilibrium that is most efficient.

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Coordination Games and
Continuous Strategy Spaces
gy p
More Complicated Simultaneous
Move Games
Other Coordination Games
• Suppose you and a partner are asked to choose one element
from the following sets of choices. If you both make the same
choice, you earn $1, otherwise nothing.
– {Red, Green, Blue}
– {Heads, Tails}
– {7 100 13 261 555}
{7, 100, 13, 261, 555}
• Write down an answer to the following questions. If your
partner writes the same answer you win $1, otherwise
nothing.
– A positive number
– A month of the year
– A woman’s name
Anti‐Coordination Games
• Not the same dress! (Fashion)
• Not the same words/ideas (Writing)
• LUPI, a Swedish lottery game designed in
2007:
200
– Choose a positive integer from 1 to 99 inclusive.
– The winner is the one person who chooses the
lowest unique positive integer (LUPI).
– If no unique integer choice, no winner.
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Another Example of Multiple Equilibria in Pure
Strategies: The Alpha‐Beta game.
• Strategies for Alpha (Row) are Up (U), Middle (M) or Down (D).
Strategies for Beta (Column) are Left (L) or Right (R).
• What are the multiple equilibria in this game? Why?
Finding Equilibria by Eliminating
Dominated Strategies....
• Strategies U and M are weakly dominated for Player Alpha by
strategy D.
• Suppose we eliminate strategy U first. The resulting game is:
• It now appears as though there is a unique equilibrium at D,R.
...Can Lead to the Wrong Conclusion.
• Suppose instead we eliminated Alpha’s M strategy first. The resulting
game is:
• Now it appears as though D,L is the unique equilibrium.
• Lesson: If there are weakly dominated strategies, consider all possible
orders for removing these strategies when searching for the Nash
equilibria of the game.
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Finding Equilibria via Best Response
Analysis Always Works:
• There are two mutual best responses: (D,L) and
(D,R).These are the two Nash equilibria of the
game.
Cournot Competition
• A game where two firms (duopoly) compete in terms of the quantity sold
(market share) of a homogeneous good is referred to as a Cournot game
after the French economist who first studied it.
• Let q1 and q2 be the number of units of the good that are brought to
market by firm 1 and firm 2.
• Assume the market price, P, is determined by market demand:
P a b(q
P=a‐b(q1+q2) if a>b(q1+q2), P=0 otherwise.
) if a>b(q ), P 0 otherwise.
P
a
Slope
=‐b
q1+q2
• Firm 1’s profits are (P‐c)q1 and firm 2’s profits are (P‐c)q2, where c is the
marginal cost of producing each unit of the good.
• Assume both firms seek to maximize profits.
Numerical Example, Discrete Choices
• Suppose P = 130‐(q1+q2), so a=130, b=1
• The marginal cost per unit, c=$10 for both firms.
• Suppose there are just three possible quantities
that each firm i=1,2 can choose qi = 30, 40 or 60.
• There are 3x3=9 possible profit outcomes for the
two firms.
• For example, if firm 1 chooses q1=30, and firm 2
chooses q2=60, then P=130‐(30+60)=$40.
• Firm 1’s profit is then (P‐c)q1=($40‐$10)30=$900.
• Firm 2’s profit is then (P‐c)q2=($40‐$10)60=$1800.
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Cournot Game Payoff Matrix
Firm 2
q2=30 q2=40 q2=60
1800,1800 1500, 2000 900, 1800
q1=30
2000, 1500 1600, 1600 800, 1200
Firm 1 q1=40
1800, 900 1200, 800 0, 0
q1=60
• Depicts all 9 possible profit outcomes for each firm.
Find the Nash Equilibrium
Firm 2
q2=30 q2=40 q2=60
Nash Equilibrium
1800,1800 1500, 2000 900, 1800
q1=30
2000, 1500 1600, 1600 800, 1200
Firm 1 q1=40
1800, 900 1200, 800 0, 0
q1=60
• q=60 is weakly dominated for both firms; use cell‐by‐cell
inspection to complete the search for the equilibrium.
Continuous Pure Strategies
• In many instances, the pure strategies available to players do
not consist of just 2 or 3 choices but instead consist of
(infinitely) many possibilities.
• We handle these situations by finding each player’s reaction
function, a continuous function revealing the action the player
will choose as a function of the action chosen by the other
will choose as a function of the action chosen by the other
player.
• For illustration purposes, let us consider again the two firm
Cournot quantity competition game.
• Duopoloy (two firms only) competition leads to an outcome in
between monopoly (1 firm, maximum possible profit) and
perfect competition (many firms, each earning 0 profits, p=c)
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Profit Maximization with Continuous Strategies
• Firm 1’s profit π1= (P‐c)q1=(a‐b(q1+q2)‐c) q1=(a‐bq2‐c) q1‐b(q1)2.
• Firm 2’s profit π2= (P‐c)q2=(a‐b(q1+q2)‐c) q2=(a‐bq1‐c) q2‐b(q2)2 .
• Both firms seek to maximize profits. We find the profit
maximizing amount using calculus:
– Firm 1: d π1 /dq1=a‐bq2‐c‐2bq1. At a maximum, d π1 /dq1=0,
• q1=(a‐bq2‐c)/2b. This is firm 1’s best response function.
(a bq c)/2b. This is firm 1 s best response function.
– Firm 2: d π2 /dq2=a‐bq1‐c‐2bq2. At a maximum, d π2 /dq2=0,
• q2=(a‐bq1‐c)/2b. This is firm 2’s best response function.
• In our numerical example, firm 1’s best response function is :
q1=(a‐bq2‐c)/2b = (130‐q2‐10)/2=60‐q2/2.
• Similarly, firm 2’s best response function in our example is:
q2=(a‐bq1‐c)/2b = (130‐q1‐10)/2=60‐q1/2.
Equilibrium with Continuous Strategies
• Equilibrium can be found algebraically or graphically.
• Algebraically: q1=60‐q2/2 and q2=60‐q1/2, so substitute out using
one of these equations: q1=60‐(60‐q1/2)/2 =60‐30+q1/4, so q1 (1‐
1/4)=30, q1=30/.75=40.
• Similarly, you can show that q2=40 as well (the problem is
perfectly symmetric).
perfectly symmetric)
• Graphically:
q1
120
60‐q1/2
40
60‐q2/2
40 120 q2
Multiple Equilibria:
Fact of Life or Problem to be Resolved?
• Suppose we have multiple Nash equilibria –the Alpha‐Beta game is an example. What
can we say about the behavior of players in such games?
• One answer is we can say nothing: both equilibria are mutual best responses , so what
we have is a coordination problem as to which equilibrium players will select. Such
coordination problems seem endemic to lots of interesting strategic environments.
– For instance there are two ways to drive, on the right and on the left and no amount of theorizing has led to
the conclusion that one way to drive is better than another. The U.S. and France drive on the right; the U.K
and Japan drive on the left.
• A second answer is that if economics strives to be a predictive science, then multiplicity of
equilibria is a problem that has to be dealt with. For those with this view, the solution is
to adopt some additional criteria, beyond mutual best response, for selecting from
among multiple equilibria.
• Some criteria used are:
– Focalness/salience (we have already seen this in the coordination games).
– Fairness /envy‐freeness (we have already seen this in the alpha‐beta game).
– Efficiency /Payoff dominance
– Risk dominance
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Efficiency
• In selecting from among multiple equilibria,
economists often make use of efficiency
considerations: which equilibrium is most efficient?
• An equilibrium is efficient if there exists no other
equilibrium in which at least one player earns a
higher payoff and no player earns a lower payoff.
• Efficiency considerations require that all players know
all payoffs and believe that all other player value
efficiency as a selection criterion.
– This may be unrealistic: Consider the Alpha‐Beta game, for
example: many pairs coordinate on D,R over D,L.
• Efficiency may be more relevant as a selection
criterion if agents can communicate/collude with one
another. Why?
Line Formation as an Example
• There are two ways for a firm to line customers up
to conduct business with spatially separated agents
of the firm:
1. There is a line for every agent of the firm, and each
customer chooses which line to get in (as at the grocery
store or a toll plaza).
store or a toll plaza)
or
2. There is one “snake line,” and customers at the head of
the line go to the first agent who becomes available (as at
a bank or airport check‐in counter).
• Efficiency considerations govern the choice between
these two line conventions. Can you explain why?
Efficiency Considerations Cannot Always be
Used to Select an Equilibrium
• There can be multiple efficient equilibria.
• Example 1: Driving on the left (U.K.) or right hand
side (U.S.) of the road.
• Example 2: The Pittsburgh Left‐Turn Game.
– Consider 2 cars who have stopped at an intersection
opposite one another. Driver 1 is signaling with the turn
i h Di 1 i i li ih h
signal that she plans to make a left turn. The other driver,
Driver 2, by not signaling indicates his plan is to proceed
through the intersection.
– The light turns green. Drivers 1 and 2 have to
simultaneously decide whether to proceed with their
plans or to yield to the other driver.
– What are the payoffs to each strategy?
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The Pittsburgh Left‐Turn Game
Driver 1
Driver 2
Payoffs are in terms of Driver 2 (Plan: proceed
seconds gained or lost
d i d l t through intersection)
th hi t ti )
(with minus sign) Proceed with Yield to
Plan Other Driver
Proceed with -1490, -1490 5, -5
Driver 1 Plan
(Plan: make Yield to -5, 5 -10, -10
left turn) Other Driver
Equilibria in the Pittsburgh Left‐Turn Game
• Both equilibria have one driver proceeding, while the other yields.
• Both equilibria are efficient.
• In such cases, governments often step in to impose one
equilibrium by law.
This game is a particular Driver 2 (Plan: proceed
version of the game of
i f th f through intersection)
th hi t ti )
“Chicken” Proceed with Yield to
Plan Other Driver
Proceed with -1490, -1490 5, -5
Driver 1 Plan
(Plan: make Yield to -5, 5 -10, -10
left turn) Other Driver
Risk Considerations and Equilibrium Selection.
• Consider the following game
X Y
X 75, 75 25, 60
Y 60, 25 60, 60
• What are the Nash equilibria?
• Which equilibrium do you think is the most likely one to be chosen?
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Computer Screen View
Risk Dominance
• The two equilibria are X,X and Y,Y.
• X,X is efficient (also known as payoff dominant) but Y,Y is less risky, or
what we call the risk dominant equilibrium – why?
X Y
X 75, 75 25, 60
Y 60, 25 60, 60
• In choosing Y, each player insures himself a payoff of 60 regardless of
what the other player does. So, Y is a risk‐free or “safe” strategy.
• How can we evaluate the expected payoff from playing X?
The Principle of Insufficient Reason
• If you are completely ignorant of which of n possible outcomes will occur,
assign probability 1/n that each outcome will occur.
X Y
X 75, 75 25, 60
Y 60, 25 60, 60
• Applying this principle to this two state game, we assign probability ½ to our
opponent choosing X and probability ½ to our opponent choosing Y.
• The expected payoff from playing X is: ½(75)+½(25)=50.
• The expected payoff from playing Y is: ½(60)+½(60)=60.
• Since the expected payoff from playing Y, 60 is greater than the expected
payoff from playing X, 50, Y is the preferred choice. If both players reason this
way, they end up at Y,Y.
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Some Games Have No Equilibrium in Pure Strategies
• Some games have no pure strategy equilibria.
• Consider, for example, the following “tennis game” between the Williams
sisters, Serena and Venus.
• Suppose Serena is in a position to return the ball and and can choose
between a down‐the‐line (DL) passing shot or a cross‐court (CC) diagonal
shot.
• Venus (on defense) has to guess what Serena will do, and position herself
Venus (on defense) has to guess what Serena will do, and position herself
accordingly. DL positions Serena for the DL shot, and CC for the CC shot.
• Payoffs are the fraction (%) of times that each player wins the point.
Venus Williams
DL CC
Serena DL 50, 50 80, 20
Williams CC 90, 10 20, 80
In such cases, playing a pure strategy is
usually not a winning strategy
• Using best response/cell‐by‐cell inspection, we see that
there is no pure strategy Nash equilibrium; Serena’s best
response arrows do not point to the same cell as Venus’s
best response arrows.
• I
In such cases, it is better to behave unpredictably using a
h i i b b h di bl i
mixed strategy.
Venus Williams
DL CC
Serena DL 50, 50 80, 20
Williams CC 90, 10 20, 80
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