This document provides an overview of game theory concepts. It defines key terms like games, strategies, payoffs, optimal strategies, and payoff matrices. It discusses different types of games including zero-sum games, positive-sum games, negative-sum games, games with dominant strategies, and Nash equilibria. Specific examples analyzed include the prisoners' dilemma, the battle of the sexes, and mixed strategy equilibria. Repeated games and how they can be used to enforce cartels are also covered. The document concludes with a discussion of sequential games and how they relate to entry deterrence strategies by incumbent firms.

1. CHAPTER 5: GAME THEORY
1

2. 5.1 Introduction
 Perf. compn  firms are P takers; in monop. no compn;
 In other mkts firms make b/ral strategic interaction;
 The oligopoly theories we discussed so far are the
classical theory of strategic interaction among firms;
 In oligopoly economic agents can be studied by using
the apparatus of game theory (GT); i.e. the right tool to
examine strategic b/n in econ. circumstances is GT, the
study of how people play games;
 GT is concerned with the general analysis of strategic
interaction of agents/players;
2

3. Introduction … Cont’d
Defn of Terms in GT:
 A Game: is any situation in which players (participants)
make strategic decisions – i.e. decisions that take in to
account each other’s actions & responses;  objective
of a game is to determine optimal strategy.
 Strategy: is rule or plan of action for playing the game
or specific course of action with clearly defined policy.
 Payoffs: outcomes that generate rewards or benefits,
or it is the net gain it will bring to the player for any
given counterstrategy of the competitor;
  Optimal strategy: is the one that maximizes player’s
expected payoff
3

4. Introduction … Cont’d
 Payoff matrix: is a table showing the payoffs accruing
to the player as a result of each possible combination of
strategies adopted by her/him & by her/his rivals;
 Players of a game: participants in a game
 Zero-sum-game: gain earned by one player is exactly
equal in magnitude to the loss incurred by opponent;
 Positive-sum-game: gain received by one of the player
is necessarily greater than loss incurred by other player;
 Negative-sum-game: loss incurred by one is necessarily
less than earning by the other;
4

5. 5.2 Dominant Strategy (DS) and Nash Eqbm
 DS refers to the optimal choice for a player no matter
what the opponent does;
 The DS maximizes the expected payoff of a player no
matter what the other does;
 E.g. Firms: A & B sell competing products  decision to
advertise (one affected by other’s decision);
 A acts on adv’t no matter what B does, & vice versa;
5
Firm B
Advertise Don’t adv
Firm A
Advertise 12, 5 15, 2
Don’t adv 7, 9 8, 3
Table 5.1 Dominant
strategy eqbm

6.   Adv’t is DS for both firms
 When every player has a DS, the outcome of the game
is equilibrium in dominant strategies;
 Do all games have DS for all players?
 What is the logical outcome of the game if payoff for A
in the bottom-right corner of the matrix is changed to
16?
 Nash Equilibrium (NE)?
 In games many one or more players may not have DS;
 See table below!
 When B chooses Left, A opts to choose for Top; 6
Dominant followed by NE … Cont’d

7.  However, when B chooses Right, A would want to
choose bottom;
 A’s optimal choice depends on what she/he thinks
about B; likewise, B’s optimal depends on what
she/he thinks about A;
 Eliminating dominated strategies doesn’t work;
 NE: each player chooses strategy that maximizes their
expected payoff, given strategies employed by others;
7
Nash Eqbm … Cont’d
Firm B
Left Right
Firm A
Top 3, 2 1,1
Bottom 1, 1 2, 3
Table 5.2 No dominant
strategy

8.  Once again:
 DS: I’m doing best I can no matter what you do, & you are
doing the best you can no matter what you are doing;
 NE: I’m doing the best I can, given what you are doing, & you
are doing the best you can, given what I am doing;
 DS eqbm is a special case of NE
 NE is justified as a solution for games b/c of
 If players are playing NE, no one has an incentive to change
their play or re-think their strategy (it is stable outcome);
 Other potential outcomes don’t have that property: if an
outcome is not NE, at least one have an incentive to change;
8
Nash Eqbm … Cont’d

9.  Does a game always has unique NE?
 Let’s see the battle of the sexes (Chicks’ Conflict):
 Married couple who are going to meet each other after
work, but haven’t decided where they are meeting;
 Let their option be baseball or ballet;
 Both prefer to be with each other, but the man prefers
the baseball game & she prefers the ballet;
9
NE followed by Game Battlers … Cont’d
Woman
Baseball Ballet
Man
Baseball 3, 2 1, 1
Ballet 0, 0 2, 3
Table 5.3 The battle of
the sexes

10.  The man would prefer that both go to the baseball
game, and the woman that both go to the ballet;
 So, how do they compromise?
 Cooperation is optimal solution;
 That is,
 They each get 2 payoff pts for being with each other, and an
additional pt for being at their preferred entertainment;
 Iterated elimination of dominated strategies eliminates
nothing;
 There are 2 Nash equilibria: one in w/c they both go to
the baseball game, and one in w/c both go to ballet;
10
Game Battlers … Cont’d

11. 5.3 The Prisoners’ Dilemma (PrD)
 One of the most widely used examples for DS eqbm;
 In, PrD two criminals named prisoner A and prisoner B
have been detained by police & questioned separately;
 They are jointly guilty of participating in a crime;
 The problem is solved eliminating dominated strategies
for each player step by step until we find the eqbm in DSs;
 If only one confesses, that prisoner serves only 1 yr in jail
while the other detained for 20 yrs;
11
Prisoner B
Confess Don’t
Prisoner B
Confess (-10, -10) (-1, -20)
Don’t (-20, -1) (-2, -2)
Table 5.4 Prisoners’
dilemma

12.  E.g.
 If B confesses & A does not, A will be imprisoned for 20 yrs,
and B loses 1 yrs only otherwise be free;
 If both confess, they are convicted any & neither goes free,
but they only serve 10 yrs each;
 If neither confesses, there is a chance they are convicted any
way (using evidence other than the confession);
 Hence, A has a strict advantage to confessing, no
matter what B is going to do;  PrD is easily solvable;
 One will be better off from confessing (DS eqbm);
presence of DS makes the PrD readily easy to solve;
12
Prisoners in dilemma … Cont’d

13.  It is unlike that of Pure Strategy (PrS) where each agent
make one choice & stick to it;
 Players not stick to a given strategy & prefers to play
random decisions or strategies;
 E.g. in matching the pennies where there is no Nash
eqbm, players randomize b/c it is unpredictable what
rivals to play;
 Random strategies based on set of chosen probabilities
 mixed strategy (MxS);
 Although there is no NE in PrSs, it exists in MxSs;
13
5.4 Mixed Strategy Equilibrium

14.  In table below, A might play Top 50% of the time &
bottom 50% of the time, while B might choose to play
left 50% of the time & right 50% of the time  MxS;
 If A follows MxS, then they have the probability of ¼ of
ending up in each of the four cells in the payoff matrix;
 Average payoff for A will be
0 [= (1/4*0) + (1/4*0) +(1/4*1) + (1/4*-1)];
 Similarly, the average payoff to B will be ½.
14
Mixed Strategy … Cont’d
Player B
Left Right
Player A
Top 0, 0 0, -1
Bottom 1, 0 -1, 3
Table 5.5 No NE in pure
strategies

15.  A NE in MxSs refers to an eqbm in w/c each agent
chooses the optimal frequency with w/c to play his/her
strategies given the frequency choices of the other;
 A MxS NE involves at last one playing a randomized
strategy, and no player being able to se their expected
payoff by playing an alternative strategy;
 MxS NE is a NE in the sense that neither party can
improve their payoff given the b/r of the other party;
 Consider matching pennies where there is no NE at all;
 Suppose row believes column plays Heads with prob. p;
then row plays heads, it gets 1 with probability p and -1
with prob. (1-p), for expected value of 2p-1;
15
Mixed Strategy … Cont’d

16.  Similarly, if row plays tails, Row gets -1 with probability
P (when Column plays Heads), & 1 with prob. (1-p), for
an expected value of 1-2p; so on for column player.
 If 2p-1 > 1-2p, Row is better off on average playing
Heads than tails, so on.
16
Mixed Strategy … Cont’d
Column player
Head Tail
Row’s expected
payoff
Row
player
Head (1, -1) (-1, 1)
1P + -1(1 - P) =
2- 1
Tail (-1, 1) (1, -1)
-1P + 1(1 - P) =
1 – 2P
Column’s
expected payoff
-1q + 1(1 - q)
= 1 – 2q
1q + -1(1 - q)
= 2q- 1
Table
5.6
expected
payoff
of
players
in
matching
the
pennies

17.  In previous notions, the game is played only once;
 Do result of a game change due repetition?
 The application of prisoners’ dilemma in to monopoly
and oligopolistic mkts:
 If both make high P (HP), both earn high profit;
 One afraid to charge HP b/c if competitor charges LP it
will lose money & its competitor get rich at its expense;
17
5.5 Repeated Game and Enforcing a Cartel
Firm 1
Low price (LP) High price (HP)
Firm 2
Low price -10, -10 0, -20
High price 20, 0 -1, -1
Table 5.7 The PrD b/n
oligopolistic firms

18.   they both choose to play safe by charging LP & final
eqbm will be (LP, LP) w/c is not as interesting as (HP, HP);
 However firms set P & output decisions over & over
again  repeated games (RG);
 In a RG each player has the opportunity to establish a
reputn for coopn, & encourage the other to do same;
 Viability of the strategy depends on whether the game
is played a fixed no of times or an infinite no of times;
 Finite No of Repetition:
 Players cooperate b/c they hope that coopn induce further
coopn in the future;
 But this requires that there will be possibility of future play;
18
Repeated Game … Cont’d

19.  Since there is no possibility of future play in the last round,
no one cooperate;
 That is, if the game has a known, fixed no of rounds, each
player will prefer not to cooperate & charge LP every round;
 If there is no way to enforce coopn on the last round, there
will be no way to enforce coopn on the next to the last round.
 Infinite No of Repetition:
 In repetition of an indefinite no of times, we do have a way of
influencing our opponent’s b/r;
 The threat of non-coopn may be sufficient to convince people
to play the Pareto efficient strategy: coopn;
 Tit-for-tat strategy (TTS): when situations characterized as
PrD – TTS offers immediate punishment for non-coopn.
19
Repeated Game … Cont’d

20.  Enforcing a Cartel:
 Both collusive & non-collusive oligopoly mkt can described by
RG b/c price & output decisions are made every now & then;
  the pricing strategy has the same structure as the PrD;
 The NE occur when each charge the lowest possible price;
 If one cuts its price in one period, others would retaliate on
the next period – follow TTS;
 The threat implicit in tit-for-tat allow firms to maintain HP or
may make members of a cartel to stick to the agreed upon P
or output level;
 Cheaters are punished!
20
Repeated Game … Cont’d

21.  In some games players move sequentially rather than
simultaneously (one moves 1st & the other responds);
 What type of equilibria would sequential games entail?
 Examples:
 One firm sets output before the other does;
 Advertising decision by a firm & the competitor's response;
 Entry-deterring inv’t by an incumbent firm & the decision
whether to enter the mkt by a potential competitor;
 New gov’t regulatory policy & the inv’t & output response of
the regulated firms.
21
5.6 Sequential Games

22.  Easier game: think of possible action & rational rxn;
 In table below, in the 1st round, player A gets to choose
top or bottom; B gets to observe the 1st player’s choice
& then choose left or right;
 Two NE: (top, left) and (bottom, right);
 But one of these equilibria is not really reasonable;
 B/c the matrix hides that one player gets to know what
the other player has chosen before making its choice;
22
Sequential Games … Cont’d
Player B
Left Right
Player A
Top 1, 9 1, 9
Bottom 0, 0 2, 1
Table 5.8 Payoff matrix
of a sequential game

23.   SGs are easier to visualize when presented in form of
decision tree  extensive form of a game – a way to
represent the game that shows time pattern of choices;
 When B makes its choice, it knows what A has done;
 If A has chosen top, it does not matter what B does, &
the payoff is (1, 9); if A has chosen bottom, the sensible
thing for B to do is to choose right, (2, 1).
23
Sequential Games … Cont’d
Top
Bottom
Player
B
Player A
B
Left
Right
Left
Right
(1, 9)
(1, 9)
(0, 0)
(2, 1)
Table 5.9 A game
in extensive form

24.  But the reasonable thing for B is to choose bottom 
the eqbm choice will be (bottom, right);  (top, left)
strategies are not a reasonable eqbm b/c it is silly for A
to ever choose top!
 But B can still threaten to play left if A plays bottom;
 Thus A might be advised to play top b/c it would be
better to earn one rather than zero;
 Is this threat credible? If B is instructed to do so, it does
better for itself by limiting its decisions;
 Once A make its choice, it expects that player B to do
the rational thing;
24
Sequential Games … Cont’d

25.  Source of monopoly power or  include economies of
scale, patents & licenses, ownership of strategic inputs,
exclusive knolowdge of a prodn technique, etc;
 However, firms themselves can sometimes deter entry
of potential competitors;
 To deter entry, the incumbent/existing firm must convince
any potential competitor that entry will be unprofitable;
 Entrant’s strategies are to enter or stay out while the
incumbent’s are either to accommodate the entrant
(maintain HP in the hope that entrant do same) or wage
warfare ( charge LP to make entry unprofitable);
25
5.7 A Game of Entry Deterrence

26.  If it is accommodating, the incumbent will earn only Br
100 million s since it has to share the mkt;
 However, if it successfully manages to deter entry and
maintain its higher P, it gets Br 200m;
 Incumbent can se its prodn capacity, produce more, &
lower its P – engage itself in P war  the entrant face
loss of 10m;
26
Entry Deterrence … Cont’d
Potential entrant
Enter Stay out
Incumbent
HP (accommodation) 100, 20 200, 0
LP (warfare) 70, -10 130, 0
Table 5.10 Entry
deterrence

27.  If entrant stays out in this case (LP), incumbent’s net
benefit will be 130m; but this does not make sense;
 Suppose incumbent threaten to expand output & start
P war in order to keep out X;
 If X takes threat seriously, it doesn’t enter b/c -10m; but
the threat is not credible if incumbent rationally acts;
 However, if incumbent makes an irrevocable com’t that
will alter incentive once entry occurs, it invest in extra
capacity needed to se output; engage in competitive
warfare should entry occur;
 If incumbent maintain HP, whether or not X enters, added
cost will reduce its payoff;
27
Entry Deterrence … Cont’d

28.  In this case, the incumbent’s threat to engage in
warfare is completely credible, &  earn high  150m;
 Short-term loses from warfare might be outweighed by
long-term gain from preventing entry in RGs;
 By fostering an image of irrationality & violence, an
incumbent firm might convince entrant that the risk of
warfare is too high;
28
Entry Deterrence … Cont’d
Potential entrant
Enter Stay out
Incumbent
HP (accommodation) 50, 20 150, 0
LP (warfare) 70, -10 130, 0
Table 5.11 Entry
deterrence – credible
threat