Game Theory with Applications to Marketing Strategy
The Rules of the Game
‧ Definitions:
 Players are the individuals who make decisions. Each player’s goal is to
maximize his utility by choice of actions.

Nature is a pseudo-player who takes random actions at specified points in
the game with specified probabilities.


An action or move by player i, denoted ai, is a choice he can make.
Player i’s action set, Ai = {ai}, is the entire set of actions available to him.

An action combination is an ordered set a = {ai} (i = 1, . . . , n) of one
action for each of the n players in the game.

Player i’s strategy si is a rule that tells him which action to choose at each
instant of the game, given his information set.

Player i’s strategy set or strategy space Si = {si} is the set of strategies
available to him.

A strategy combination s = (s1, . . . , sn) is an ordered set consisting of one
strategy for each of the n players in the game.

By player i’s payoff πi(s1, . . . , sn), we mean either:
(1) The utility player i receives after all players and Nature have picked
their strategies and the game has been played out; or
(2) The expected utility he receives as a function of the strategies chosen by
himself and the other players.
‧ Extensive form
 A configuration of nodes and branches running without any closed loops
from a single starting node to its end nodes.
 An indication of which node belongs to which player.
 The probabilities that Nature uses to choose different branches at its nodes.
 The information sets into which each player's nodes are divided.

The payoffs for each player at each end node.
‧ Definitions:
 A node is a point in the game at which some player or Nature takes an
action, or the game ends.


A branch is one action in a player's action set at a particular node.
Player i's information set at any particular point of the game is the set of
different nodes in the game tree that he knows might be the actual node, but
between which he cannot distinguish by direct observation.
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Figure 1
‧ Normal form
 A list of players
 For each player, a list of strategies

For each array of strategies, one for each player, a list of payoffs that the
players receive.
‧ Write the games depicted in figure 1~3 in normal form.
Figure 2
Figure 3
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‧ Consider the extensive form game depicted in figure 4.
 What are player 2’s information sets? What are player 1’s?
 Write this game in normal form.
Figure 4
‧ Best response: Player i’s best response or best reply to the strategies s-i chosen
by the other players is the strategy si* that yields him the greatest payoff; that is
 i ( si* , s i )   i ( si' , si ) si'  si* .
‧ Dominant strategy: The strategy si* is a dominant strategy if it is a player’s
strictly best response to any strategies the other players might pick, in the sense
that whatever strategies they pick, his payoff is highest with si* . Mathematically,
 i ( si* , s i )   i ( si' , si )
s i , si'  si* .
‧ Dominated strategy: The strategy sid is a dominated strategy if it is strictly
inferior to some other strategy no matter what strategies the other players choose,
in the sense that whatever strategies they pick, his payoff is lower with sid .
Mathematically, sid is dominated if there exists a single si' such that
 i ( sid , si )   i ( si' , si )
s i .
‧ A dominant-strategy equilibrium is a strategy profile consisting of each
player’s dominant strategy.
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‧ Prisoner’s Dilemma
 P. 12~14
 Examples in marketing
 The frequent-flier program
 How can the firms escape from the prisoner’s dilemma?
‧ A seller has one indivisible unit of an object for sale. The n bidders have
valuations 0  v1  v2  ...  vn for the object, and these valuations are common
knowledge. Bidders submit bids bi simultaneously. The highest bidder wins the
object and pays the second bid; i.e., if i wins, he has utility
 i  vi  max b j .
j i
The others do not pay anything.
(i) Show that bidding one’s valuation (bi = vi) is a dominant strategy for bidder i.
(ii) Conclude that bidder n wins and has a surplus of vn-vn-1.
(iii) Would these results be affected if each trader knew his own valuation but not
those of the other bidders?
‧ Strategy si' is weakly dominated if there exists some other strategy si'' for
player i which is possibly better and never worse, yielding a higher payoff in some
strategy profile and never yielding a lower payoff. Mathematically, si' is weakly
dominated if there exists si'' such that
 i ( si' , s i )   i ( si'' , si )
s i , and
 i ( si' , s i )   i ( si'' , s i ) for some s i .
Similarly, we call a strategy that is always at least as good as every other
strategy and better than some a weakly dominant strategy.
‧ An iterated-dominance equilibrium is a strategy profile found by deleting a
weakly dominated strategy from the strategy set of one of the players,
recalculating to find which remaining strategies are weakly dominated, deleting
one of them, and continuing the process until only one strategy remains for each
player.
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‧
Player 1 / Player 2
L
M
R
U
0, -1
0, 0
1, 1
M
2, 3
3, 1
3/2, -1
D
4, 2
1, 1
2, 3/2
‧ Consider the Cournot game. The set of players: two firms, i =1, 2. The strategies:
two firms’ output quantities: qi  [0, ) , i =1, 2. The payoff of each player i:
 i ( qi , q j )  qi [ P( qi  q j )  c]  F ,
where P(*) is the inverse demand function, c and F are respectively the variable
and fixed costs. Suppose that c = F = 0 and P = 1 – q1 – q2. Find out an
iterated-dominance equilibrium.
‧ Two firms are competing in a declining industry in continuous time. If at time
t  [0, ) both stay, firm i gets profit  id ( t ) ; if only firm i stays, it gets  im (t ) .
It is given that
 1m = 11/4 – t,
 2m =7/4 – t,
 1d = 2 – 2t,
 2d = 1 – 2t.
‧ Market share competition: In the below, two firms can spend c > 0 on promotion.
One firm would capture the entire market by spending c if the other firm does not
do so. Show that when c is sufficiently small, this game has a dominance
equilibrium.
Player 1 / Player 2
Don’t Promote
Promote
Don’t Promote
1, 1
0, 2-c
Promote
2-c, 0
1-c, 1-c
‧ Consider also the following moral-hazard-in-term problem: Two workers can
either work (s = 1) or shirk (s = 0). They share the output 4(s1+s2) equally.
Working however incurs a private cost of 3. Show that this game has a dominance
equilibrium.
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