Game Theory
Learning Objectives
• Define game theory, and explain how it helps
to better understand mutually interdependent
management decisions
• Explain the essential dilemma faced by
participants in the game called Prisoners’
Dilemma
• Explain the concept of a dominant strategy
and its role in understanding how auctions can
help improve the price for sellers, while still
benefiting buyers
Overview
I.
II.
III.
IV.
V.
Introduction to Game Theory
Simultaneous-Move, One-Shot Games
Infinitely Repeated Games
Finitely Repeated Games
Multistage Games
Game Theory
• Optimization has two shortcomings when applied to
actual business situations
– Assumes factors such as reaction of competitors or
tastes and preferences of consumers remain constant.
– Managers sometimes make decisions when other
parties have more information about market
conditions.
• Game theory is concerned with “how individuals make
decisions when they are aware that their actions affect
each other and when each individual takes this into
account.”
• Game Theory is a useful tool for managers
• In the analysis of games, the order in which
players make decisions is important
• Simultaneous-move game- Each player makes
decision without knowledge of other players
decision
• Sequential-move game: player makes a move
after observing other player’s move
• One shot game – underlying game is played only
once
• Repeated game – underlying game is played
more than once
• How managers use game theory:
Betrand Duopoly game:
2 gas stations – no location advantage.
Consumers view product as perfect substitutes
and will purchase from station that sells at
lower price.
First thing manager must do in the morning is to
tell attendant to put up price without
knowledge of rival’s price.
This is a simultaneous move game.
If Manager of station A calls in price higher than
B  will lose sales that day
Normal Form Game
• A Normal Form Game consists of:
– Players.
– Strategies or feasible actions.
– Payoffs.
A Normal Form Game
Player 1
Player 2
Strategy
a
b
c
A
B
C
12,11
11,10
10,15
11,12
10,11
10,13
14,13
12,12
13,14
Simultaneous-move, One shot
game
• Important to managers making decisions in an
environment of interdependence. E.g. profits
of firm A depends not only on firm’s A actions
but on the actions of rival firm B as well.
Normal Form Game:
Scenario Analysis
Player 1
Player 2
Strategy
Up
Down
Left
Right
10,20
15,8
10,10
-10, 7
• What’s the optimal strategy?
Complex question. Depends on the nature
game being played.
The game above is easy to characterize the
optimal decision– a situation that involves a
dominant strategy.
A strategy is dominant if it results in the highest
payoff regardless of the action of the
opponent
• For player 1, the dominant strategy is UP.
Regardless of what player 2 chooses, if A
chooses UP, she’ll earn more.
• Principle:
Check to see if you have a dominant strategy. If
you have one, play it.
What should a player do in the absence of a
dominant strategy (e.g. Player 2)?
Play a SECURE STRATEGY
-- A strategy that guarantees the highest payoff
given the worst possible scenario.
Find the worse payoff that could arise for each
action and choose the action that has the
highest of the worse payoffs.
Secure strategy for player 2 is RIGHT.
Guarantees a payment of 8 rather than 7 from
LEFT
2 shortcomings:
1. Very conservative strategy
2. Does not take into account the optimal
decision of your rival and thus may prevent
you from earning a significantly higher
payoff.
Player 2 should actually choose LEFT,
knowing that player 1 will play UP
Principle: Put yourself in your rival’s shoes
If you do not have a dominant strategy, look at
the game from your rival’s perspective. If your
rival has a dominant strategy, anticipate that
she will play it.
Putting Yourself in your Rival’s Shoes
• What should player 2 do?
– 2 has no dominant strategy!
– But 2 should reason that 1 will play “a”.
– Therefore 2 should choose “C”.
Player 1
Player 2
Strategy
a
b
c
A
B
C
12,11
11,10
10,15
11,12
10,11
10,13
14,13
12,12
13,14
The Outcome
Player 1
Player 2
Strategy
a
b
c
A
B
C
12,11
11,10
10,15
11,12
14,13
10,11
10,13
12,12
13,14
• This outcome is called a Nash equilibrium:
– “a” is player 1’s best response to “C”.
– “C” is player 2’s best response to “a”.
Nash Equilibrium
• Given the strategies of other players, no
player can improve her payoff by unilaterally
changing her own strategy.
• Every player is doing the best she can given
what other players are doing.
• In original example, Nash equilibrium is when
A chooses UP and B chooses LEFT.
Application of One shot games
• Two managers want to maximize market
share.
• Strategies are pricing decisions. (charge high
or low prices)
• Simultaneous moves.
• One-shot game. (firms meet once and only
once in the market)
The Market-Share Game
in Normal Form
Manager 1
Manager 2
Strategy Low Price High price
Low Price
0, 0
50, -10
High Price -10,50
10,10
Market Share game Equilibrium
• Each manager’s best decision is to charge a
low price regardless of the other’s decision.
Outcome of game is that both firms charge a
low price and earn 0 profits
• Low prices for both managers is the Nash
Equilibrium
• If firms collude to charge high prices, profits
will be higher for both
•  Classic case in Economics called dilemma
because the Nash equilibrium outcome is
inferior (from the firms viewpoint) to the
situation where they both “agree” to charge
high prices
Even if firms meet secretly to collude, is there an
incentive to “cheat” on the agreement?
To advertise or Not?
• Your firm competes against another firm for
customers
• You and your rivals know your product will be
obsolete at the end of the year (one shot
game) and must simultaneously determine
whether or not to advertise.
• In your industry, advertising does not increase
industry demand but induces consumers to
switch among the products of the different
firms
An Advertising Game
Manager 1
Manager 2
Strategy Advertise
Advertise
4,4
No Ad
1,20
No Ad
20,1
10,10
To advertise or Not?
• Dominant strategy of each firm is to advertise.
 unique Nash equilibrium.
• Collusion will not work because this is a oneshot game and if there’s agreement not to
advertise, each firm will have an incentve to
cheat.
Key Insight:
• Game theory can be used to analyze
situations where “payoffs” are non
monetary!
• We will, without loss of generality, focus on
environments where businesses want to
maximize profits.
– Hence, payoffs are measured in monetary
units.
Examples of Coordination Games
• Industry standards
– size of floppy disks.
– size of CDs.
• National standards
– electric current.
– traffic laws.
• Coordination Decisions:
Firms don’t have competing objectives but
coordinating their decisions will lead to higher
profits
e.g. Producing appliances that require either 90volt or 120-volt outlets
A Coordination Game in Normal
Form
Firm A
Firm B
Strategy
120-volt
90-volt
120-volt
100,100
0,0
90-volt
0,0
100,100
Coordination Game: 2 Nash Equilibria
• What would you do if you manage Firm A?
If you do not know what firm B is going to do,
you’ll have to guess what B will do.
Effectively, both you and firm B will do better by
coordinating your actions.
2 Nash equilibria. If the firms can ‘talk’ to each
other, they can agree on what to produce.
Notice, there’s no incentive to cheat here
This is a game of coordination rather than game
of conflicting interest
Simultaneous-Move Bargaining
• Management and a union are negotiating a wage
increase.
• Strategies are wage offers & wage demands.
• Players have one chance to reach an agreement and
offer is made simultaneously.
• Parties are bargaining over how much of $100 in
surplus must go to the union
• Assume the surplus can be split only into $50
increments
• One shot to reach agreement
• Parties simultaneously write the amount they desire
on a piece of paper.
• If the sum of the amounts does not exceed $100,
players get the specified amount
• If sum exceeds $100, stalemate, costing each player $1
The Bargaining Game
in Normal Form
Management
Union
Strategy
0
50
100
0
0,0
50,0
100,0
50
0,50
50,50
-1,-1
100
0,100
-1,-1
-1,-1
Simultaneous-Move Bargaining
• 3 Nash equilibria outcomes.
• Multiplicity of equilbria leads to inefficiency if parties
fail to “co-odinate” on an equilibrium
• 6 of 9 outcomes are inefficient because they don’t sum
up to 100
• Clearly, in this game management must ask for 50 if
they
Key Insights:
• Not all games are games of conflict.
• Communication can help solve coordination
problems.
• Sequential moves can help solve
coordination problems.
Infinitely Repeated Games
• Game played over and over again. Players
receive payoff during each repetition of game
• Firms compete week after week, year after
year  game is repeated over time
• To evaluate profits earned during this game,
consider the PV of all payoffs.
• If payoffs are the same in each period, then
for an infinitely played game
• PV = (1+i)/i * constant profit
An Advertising Game
• Two firms (Kellogg’s & General Mills)
managers want to maximize profits.
• Strategies consist of pricing actions.
• Simultaneous moves.
– Repeated interaction.
Equilibrium to the One-Shot
Pricing Game
Kellogg’s
General Mills
Strategy
Low
High
Low
0,0
-40,50
High
50, -40
10,10
• When firms repeatedly face this type of
matrix, they use “trigger strategy”
• Trigger Strategy – is a strategy that is
contingent on the past plays of players in a
game
• A player who adopts a trigger strategy
continues to choose the same action until
some other player takes an action that
“triggers” a different action by the first player
Can collusion work if firms play the
game each year, forever?
• Consider the following “trigger strategy” by
each firm:
– “We will each charge the high price, provided
neither of us has ever “cheated” in the past. If
one of us cheats and charges a low price, the
other player will “punish” the deviator by charging
low price in ever period thereafter”
• In effect, each firm agrees to “cooperate” so
long as the rival hasn’t “cheated” in the past.
“Cheating” triggers punishment in all future
periods.
Kellogg’s profits?
Cooperate = 10 +10/(1+i) + 10/(1+i)2 + 10/(1+i)3 + …
= 10 + 10/i
Value of a perpetuity of $12 paid
Cheat = 50+0 +0 +0 +0
at the end of every year
There’s no incentive to cheat if the PV from cheating is less than
the PV from not cheating
Kellogg’s Gain to Cheating:
• Cheat - Cooperate = 50 - (10 + 10/i) = 40 - 10/i
– Suppose i = .05
• Cheat - Cooperate = 40- 10/.05 = 40 - 200 = -160
• It doesn’t pay to deviate.
• As long as i is less than 25%, it pays not cheat.
– Collusion is a Nash equilibrium in the infinitely repeated
game!
Benefits & Costs of Cheating
• Cheat - Cooperate = 40 - 10/i
– 40 = Immediate Benefit (50 - 10 today)
– 10/i = PV of Future Cost (10 - 0 forever after)
• If Immediate Benefit - PV of Future Cost > 0
– Pays to “cheat”.
• If Immediate Benefit - PV of Future Cost  0
– Doesn’t pay to “cheat”.
Application of Infinitely repeated
games (product quality)
Consumers
Firm
Strategy Low Quality
Don't Buy
0,0
Buy
-10,10
High
0, -10
1,1
• If one shot game, Nash equilibrium = low
quality product and don’t buy
• If infinitely repeated and consumers tell firm:
“I’ll buy your product and will continue to buy
if it is of good quality. But if it turns out to be
shoddy, I’ll tell my friends not to buy anything
from you again”.
• Given this strategy of consumers, what should
the firm do?
• If the interest rate is not too high, the best
alternative is to sell a high product quality
• If firm cheats and sells shoddy product, it will
earn 10 now but 0 forever thereafter.
• It will not pay for the firm to cheat if the
interest rate is low.
FINITE REPEATED GAMES
Games that eventually end
1. Games in which players do not know when
the game will end
2. Games in which players know when it will
end.
• Suppose two duopolists repeatedly play the
pricing game until their product become
obsolete. Suppose the firms don’t know when
the game will end but there’s a probability p
that the game will end after every given play
• Probability the game will be played tomorrow
if played today is (1-p). If the game is played
tomorrow, the probability it will be played the
next day is (1-p)2 etc.
Pricing Game that is infinitely
repeated
Kellogg’s
General Mills
Strategy
Low
High
Low
0,0
-40,50
High
50, -40
10,10
• Suppose firms adopt trigger strategies,
whereby each agrees to charge a high price
but if a firm deviates and charges a low price,
the other firm will punish it by charging low
price until the game ends.
• Assume interest rate is zero
• Does Kellogg’s have an incentive to cheat?
Kellogg’s profits?
Cooperate = 10 +10/(1-p) + 10/(1-p)2 + 10/(1-p)3 + …
= 10/p
Cheat = 50+0 +0 +0 +0
There’s no incentive to cheat if the profit from cheating is less
than the profit from not cheating. If there is a 10% that the
government will ban the sale of the item, then profit from not
cheating is 100  It pays not to cheat
Key Insight
• Collusion can be sustained as a Nash
equilibrium when there is no certain “end” to
a game.
• Doing so requires:
– Ability to monitor actions of rivals.
– Ability (and reputation for) punishing defectors.
– Low interest rate.
– High probability of future interaction.
End of Period Problem
• When players know precisely when a repeated
game will end, end-of-period problem arises
• In the final period, there’s no tomorrow and
there’s no way to punish a player for doing
something wrong in the last period.
• Consequently, players will behave as if it was a
one shot game
Resignations, Quits & Snake Oil salesmen
• Workers work hard if threatened with being
fired if benefits of shirking are less than cost
of being fired
• When worker announces that she wants to
quit, say tomorrow, the cost of shirking is low
so threat of firing has no effect
• What can managers do to overcome problem?
1. Fire the worker as soon as she announces
plan to quit? Problems
Snake Oil Salesmen move about so no
punishments
Factors affecting collusion in pricing
games
• Number of firms: Collusion is easier when there are
few firms rather than many.
• Firm Size: Economies of scale exists in monitoring.
Easier for large firms to monitor small ones than other
way round
• History of the Market: Explicit meeting to collude or
tacit collusion?
• Punishment Mechanism: How do we punish our rivals
when they cheat?
Real World Examples of Collusion
•
•
•
•
Garbage Collection Industry
OPEC
NASDAQ
Airlines
Multistage Games
• Timing is important
• Players make sequential rather than
simultaneous decisions
The Game in Extensive Form
UP
10, 15
DOWN
5, 5
UP
0, 0
B
UP
A
DOWN
B
DOWN
6, 20
• Player A must make a decision before player B.
A cannot make actions conditioned on what B
does
• Player B’s action is dependent on what A does
• Suppose player B’s strategy is : Choose down if
player A chooses up, and down if player A
chooses down”
• Given this, best choice for A is down and earn
6
• If A chooses down, does B have an incentive to
change her strategy? No  these two strategies
lead to a Nash equilibrium
• But this is not a subgame perfect equilibrium
because there’s a superior Nash equilibrium
• Suppose B’s strategy is choose up if A chooses up and
choose down if A chooses down, then A will choose
up, earning 10, a better position than choosing down
• In the first equilibrium, A chose down because of B’s
threats. But is the threat really credible?
Find the Subgame Perfect Nash
Equilibrium Outcomes
• Outcomes where no player has an incentive to
change its strategy, given the strategy of the
rival, and
• The outcomes are based on “credible actions;”
that is, they are not the result of “empty
threats” by the rival.
Pricing to Prevent Entry: An
Application of Game Theory
• Two firms: an incumbent and potential entrant.
• Potential entrant’s strategies:
– Enter.
– Stay Out.
• Incumbent’s strategies:
–
–
–
–
{if enter, play hard}.
{if enter, play soft}.
{if stay out, play hard}.
{if stay out, play soft}.
• Move Sequence:
– Entrant moves first. Incumbent observes entrant’s action and
selects an action.
The Pricing to Prevent Entry Game in
Extensive Form
-1, 1
Hard
Incumbent
Enter
Soft
5, 5
Entrant
Out
0, 10
Identify Nash and Subgame Perfect
Equilibria
-1, 1
Hard
Incumbent
Enter
Soft
5, 5
Entrant
Out
0, 10
Two Nash Equilibria
-1, 1
Hard
Incumbent
Enter
Soft
5, 5
Entrant
Out
0, 10
Nash Equilibria Strategies {player 1; player 2}:
{enter; If enter, play soft}
{stay out; If enter, play hard}
One Subgame Perfect Equilibrium
-1, 1
Hard
Incumbent
Enter
Soft
5, 5
Entrant
Out
0, 10
Subgame Perfect Equilibrium Strategy:
{enter; If enter, play soft}
Insights
• Establishing a reputation for being unkind to
entrants can enhance long-term profits.
• It is costly to do so in the short-term, so much
so that it isn’t optimal to do so in a one-shot
game.
Games of Particular
Relevance in Economics
• Beach Kiosk Game
– Two-Person, Zero-Sum, Non-cooperative
– Example: two companies provide snacks and
sunscreen on a beach.
• Beachgoers spread themselves out evenly along the
beach.
• Both companies ultimately locate at the midpoint of
the beach, otherwise the other company has an
advantage (closer to more beachgoers)
• Real life example: location of gas stations
Games of Particular
Relevance in Economics
• Repeated Game: game is played repeatedly
over a period of time.
• In a repeated game, equilibria that are not
stable may become stable due to the threat of
retaliation.
Games of Particular
Relevance in Economics
• Repeated Game: game is played many times, and equilibria that are
not stable may become stable due to the threat of retaliation.
• Assume (High, High) equilibrium reached and both firms start off
charging the high price.
• In the next period, if one firm cheats (charges low price), it receives
600 in that period.
• Other firm will change to low prices in the next period to “retaliate”
and both will end up at (Low, Low) equilibrium.
• Thus, incentive exists not to “cheat” in a repeated game and (High,
High) is a viable equilibrium, though it is not in a single-period
game.
• If number of periods are fixed, both firms will have incentive to
cheat (charge low price) in the last period due to lack of threat of
retaliation, which will then allow them to cheat in all periods.
Games of Particular
Relevance in Economics
• Consider the following payoff matrix in which
firms choose their capacity, either high or low.
• Suppose firm C has the ability to move first.
– C would choose Low, then D would choose High.
Game Theory and Auctions
• Non-cooperative, non-zero-sum game
• Seller wants to sell at highest price, buyer wants to
buy at lowest price.
• Dutch Auction
– All product sold at the highest price that clears the market
– Each buyer describes the quantity demanded and price to
pay
– Starting at highest price, sum quantity demanded up to
the quantity available. The associated price for the last
quantity added is the price for all products.
• In an auction with a time limit, every player has a
dominant strategy to bid as late as possible.
Strategy and Game Theory
• In Prisoners’ Dilemma, players have a dominant
strategy that leads to suboptimal results.
• Commitment, explicit or implicit, can be used to
achieve preferred outcomes.
• Commitment must be credible to have effect.
• To make a commitment credible:
– Burn bridges behind you.
– Establish and use a reputation.
– Write contracts.
• Incentives also can be used to change the game to
achieve preferred outcomes.
Strategy and Game Theory
• Fundamental aspects of game theory
– Players are interdependent
– Uncertainty: other players’ actions are not entirely
predictable
• PARTS: paradigm for studying a situation, predicting
players’ actions, making strategic decisions
– Players: Who are players and what are their goals?
– Added Value: What do the different players contribute to
the pie?
– Rules: What is the form of competition? Time structure of
the game?
– Tactics: What options are open to the players?
Commitments? Incentives?
– Scope: What are the boundaries of the game?