GAME THEORY
The essence of a game is INTERDEPENDENCE among players. Any
situation where a person or organisation takes actions that affect others,
where the others are aware of that effect and the others may act and have an
effect on the original person or organisation may be described as a game.
DEFINTIONS
 A game is distinguished from a real conflicting situation by the fact
that games are conducted by using a set of rules.
 All games have three basic elements namely: players, strategies and
payoffs.
 The decision-making units are the players. More specifically a player
could be a single person, an organisation, consumers, producers or
even nations.
 Each course of action open to a player is called a strategy.
 The rules of the game define each player’s feasible strategies. The
number of strategies available to each player is assumed to be finite.
In the Cournot model a strategy involves quantity setting and in the
price setting model the strategy centers on the price setting ability.
There are two types of strategies:
 The price strategy - this is not hidden to the other players.
 The mixed strategy – this is hidden to the other players as it attaches
probabilities to given strategies.
PAYOFFS are the returns to players of the game and such payoffs could be
in money terms, utility, output levels or profits depending on the motive of
the game. Players can rank the payoffs of the game from the most preferred
to the least preferred and will seek the highest ranked payoff attainable.
Zero sum game (also known as a constant sum game) and Non-zero sum
game (also known as a variable sum game)
 In a zero sum game the interests of the players are diametrically
opposed to each other. If one player wins amount A, the other player
loses exactly the same amount.
 In a non-zero sum game the gains and losses are not equal. Both
players may gain or lose and the gains and losses need not sum to
zero.
Sequential and Simultaneous Games
When a game is sequential each player moves in turn and each player is
aware of the moves that have been taken previously. The question each
player is trying to answer is “What should I do given what my opponent has
done and given what my opponent will do when they know that I have
moved”.
When a game is simultaneous, all players may be thought as moving at the
same time. However, timing itself is not the key feature of simultaneous
games and the moves need not take place at the same moment. What really
defines a simultaneous game is the fact that each player moves without
knowing what the other player has done. In this case the question asked is
“What should I do given that I do not know what my opponent will do and
my opponent does not know what I will do”.
Repeated and One-Off Games
Games that are repeated may be repeated an infinite number of times, a
finite and known number of times or an unknown number of times.
In a one-off game the players need only concern themselves about the gains
and losses arising from a single round. If games are repeated then each
player needs to consider the impact of their actions in each in the future.
REPRESENTING AND SOLVING GAMES
 A Tree diagram can be used to represent a game. Such a
representation specifies the order of play, the data, the choices of ply
and the payoffs to all the players.
“A” stands for decision point for player A. In the same vein “B” stands for
decision point for player B.
(7,5), 7 stands for profits or returns for firm A, and 5 stands for profit or
returns for firm B.
If A adopts L strategy then B should adopt the L strategy as well as he gets
higher profits that a case where he adopts the H strategy.
 The normal form of the game is the tabular form or matrix
representation.
Just like what we have in the first representation both players will choose the
L strategy.
DOMINANT STRATEGIES
A dominant strategy is a strategy that is optimal for a player no matter what
an opponent does. The following example illustrates this in a duopoly
setting. Suppose firms A and B sell competing products and are deciding
whether to undertake advertising campaigns. Each firm however will be
affected by its competitor’s decision. The payoff matrix shown below
illustrates the possible outcomes of the game. Recall that the payoff matrix
summarizes the possible outcomes of the game; the first number in each cell
is the payoff to firm A and the second number is the payoff to firm B.
The payoff matrix for an advertising game
Observe from this payoff matrix that if both firms decide to advertise Firm A
makes a profit of 10 and Firm B will make a profit of 5. If Firm A advertises
and Firm B does not advertise Firm A will earn 15 and Firm B will earn
zero.
What strategy should each firm choose?
First consider Firm A. Firm A should advertise no matter what Firm B does
because Firm A does best by advertising. If Firm B advertises A earns a
profit of 10 if it advertises but only 6 if it doesn’t. Thus Advertising is a
dominant strategy for Firm A.
The same is true for Firm B. No matter what Firm A does, Firm B does best
by advertising. Thus assuming that both firms are rational we know that the
outcome for this game is that BOTH FIRMS WILL ADVERTISE. This
outcome is easy to determine because both firms have dominant strategies.
However, not every game has a dominant strategy for each player. To see
this we are going to change our advertising game slightly. The payoff matrix
is the same as above except for the bottom right hand corner – if neither firm
advertises, Firm B will again earn a profit of 2 but Firm A will earn a profit
of 20.
MODIFIED ADVERTISING GAME
Now Firm A has no dominant strategy. Its optimal decision depends on what
Firm B does If Firm B advertises, then Firm A does best by advertising; but
if Firm B does not advertise Firm A also does best by not advertising.
Now suppose that both firms must make their decisions at the same
time. What should Firm A do?
To answer this Firm A must put itself in Firm B’s shoes. What decision is
best from B’s point of view, and what is Firm B likely to do? The answer is
clear: Firm B has a dominant strategy-advertise, no matter what Firm A
does. If Firm A advertises B earns 5 by advertising and zero (0) by not
advertising. If A does not advertise, B earns 8 if it advertises and 2 if it does
not. Therefore, Firm A can conclude that Firm B will advertise. This means
that Firm A should itself advertise and thereby earn 10 instead of 6.
It is the logical outcome of the game because Firm A is doing the best it can
given Firm B’s decision; and Firm B is doing the best it can given Firm A’s
decision.
THE NASH EQUILIBRIUM
Dominant strategies are stable, but in many games, one or more players do
not have a dominant strategy. We therefore need a more general equilibrium
concept.
A Nash Equilibrium is a set of strategies or actions such that each player is
doing the best he can given the actions of his opponents. Since each player
has no incentive to deviate from his Nash Strategy, the strategies are stable.
In the example above, the Nash Equilibrium is that both firms will advertise.
It is Nash Equilibrium because given the decision of its competitor, each
firm is satisfied that it has made the best decision possible and has no
incentive to change its decision.
Comparison of Nash Equilibrium and the Equilibrium of Dominant
Strategies.
DOMINANT STRATEGY EQUILIBRIUM:- I am doing the best I can no
matter what you do. You are doing the best you can no matter what I do.
NASH EQUILIBRIUM:- I am doing the best I can given what you are
doing. You are doing the best you can given what I am doing.
In the advertising game there is a single Nash Equilibrium ADVERTISE. In
general a game does not have a single Nash Equilibrium. Sometimes there is
no Nash Equilibrium, and sometimes there are several strategies that are
stable and self-enforcing.
Eg, THE PRODUCT CHOICE PROBLEM
Two breakfast cereal companies face a market in which two new variations
of cereal can each be successfully introduced by only one firm. There is a
market for a new “crispy” cereal and a market for a new “sweet” cereal, but
each firm has the resources to introduce only one new product. The payoff
matrix for the two firms is shown below.
In this game each firm is indifferent about which product it introduces, so
long as it does not introduce the same product as its competitor. If
coordination were possible, the firms would probably agree to divide the
market. So what will happen if the firms must behave
NONCOOPERATIVELY?
Suppose that somehow, perhaps through news release- Firm 1 indicates that
it is about to introduce the sweet cereal, and Firm 2 after hearing this
indicates it will introduce the crispy one. Now given the action it believes
the opponent is taking, neither firm has an incentive to deviate from its
proposed action.
If it takes the proposed action, its payoff is 10, but if it deviates given that its
opponent’s action remains unchanged, its payoff will be –5. Therefore, the
strategy set given by the bottom left-hand corner of the payoff matrix is
stable and constitutes a Nash Equilibrium. In other words given the strategy
of its opponent, each firm is doing the best it can and has no incentive to
deviate.
Note that the upper right-hand corner of the payoff matrix is also a Nash
Equilibrium, which might occur if Firm 1 indicated it was about to produce
the crispy cereal. Each Nash Equilibrium is stable because once the
strategies are chosen no player will unilaterally deviate from them.
However, without more information, we have no way of knowing which
equilibrium (crispy/sweet versus sweet/crispy) is likely to result or if either
will result. Of course both firms have a strong incentive to reach one of the
two Nash Equilibria because if they both introduce the same type of cereal,
they will lose money.
The fact that these two firms are not allowed to collude does not mean that
they will reach Nash Equilibrium.
PRISONERS’ DILEMMA GAMES
Two men are arrested for a robbery. If convicted, each will receive a jail
sentence of two to five years, the actual length depends on what the
prosecution recommends. Unfortunately for the District Attorney, he does
not have enough evidence to get a conviction.
The DA puts the criminals in separate cells. Ho goes to Joe first. He tells
him that if he confesses and Mike does not the DA will drop the burglary
charge and let Joe off with a slap on the wrist, that is, three months for
trespass. If Mike also confesses the DA cannot drop the charge but he will
ask the judge for leniency; Mike and Joe will get two years each.
If Joe refuses to confess the DA will not feel so friendly. If Mike confesses,
Joe will be convicted and the DA will ask for a maximum possible sentence.
If neither confesses the DA cannot convict them of the robbery, but will
press for a six months sentence for trespass, resisting arrest and vagrancy.
After explaining all this to Joe the DA goes to Mike’s cell and gives the
same explanation with the names reversed. Figure below shows the matrix
of outcomes facing Joe and Mike.
Joe Reasons as Follows
If Mike confesses and I don’t, I get 5 years; if I confess too I get 2 years. If
Mike is going to confess I would be better off if I confess too. If neither of
us confesses, I go to jail for six months. That is a considerable on what will
happen if Mike squeals, but I can do better; if Mike stays silent and I confess
I only get three months.
So if Mike is going to stay silent I am better off confessing. In fact whatever
Mike does I am better off confessing. Joe calls the guard and asks him to
send for the DA. It takes a while, Mike has made the same calculation and
reached the same conclusion and is in the middle of dictating his confession.
The game has at least two interesting properties. Both criminals confess
because each calculates correctly that confession is better than silence
whatever the other criminal does.
NB. Row “confess” has a higher payoff for Joe than row “say nothing”
whichever column Mike chooses. Column “confess” has a higher payoff for
Mike than column “say nothing” whichever row Joe chooses.
Confess is the dominant strategy for both criminals. The solution to the
game is therefore (2 years, 2years). The collusive equilibrium will be (6
months, 6 months).
Business Example of the Prisoners’ Dilemma
Suppose there is a symmetric version of the price war game:
So here a unique Nash Equilibrium exists where both firms cut prices (a
price war ensues). Notice a peculiar feature of this game. We can think of
the SOCIAL MAXIMUM as the set of strategies that benefits all players
(with say equal weight) the most. So the upper left corner has a Social
Benefit (SB) to the two firms (obviously not to the consumers) of SB = 3 + 3
= 6.
Similarly:
So the Nash Equilibrium is the worst possible outcome for the two
firms. In fact, if firms could only collude on prices, they would surely agree
to both hold prices high. The prisoners’ dilemma is the basis for antitrust
rules prohibiting price collusion and is perhaps the most important game in
business. By prohibiting collusion or discussion of prices, the market keeps
intact the incentive to cut prices (benefiting consumers).
MAXIMIN STRATEGIES
The concept of Nash Equilibrium relies heavily on individual rationality.
Each player’s choice of strategy depends not only on his own rationality, but
also on that of his opponent. This can be a limitation, as the example below
shows. In this game, playing “right” is the dominant strategy for Player 2
because by using this strategy, Player 2 will do better (earning 1 rather than
0), no matter what Player 1 does. Thus Player 1 should expect Player 2 to
play the “right” strategy. In this case Player 1 would do better by playing
“bottom” to earn 2 instead of playing “top” to earn 1. Clearly the outcome
(bottom, right) is the Nash Equilibrium for this game, you can verify that it
is the only Nash Equilibrium. But note that Player 1 had better be sure that
Player 2 understands the game and is rational. If Player 2 should happen to
make a mistake and play “left” it would be extremely costly to Player1.
If you were Player 1, what would you do? If you tend to be cautious, and
you are concerned that player 2 might not be fully informed or rational, you
might choose to play “top” in which case you will be assured of earning 1
and you will have no chance of losing 1000.
Such a strategy is called a maximin strategy because it maximises the
minimum gain that can be earned. If both players used maximin strategies
the outcome would be (top, right). A maximin strategy is conservative, but
not profit maximising since Player 1 earns profit of 1 instead of 2.
Note that if Player 1 knew for certain that Player 2 was using a maximin
strategy, he would prefer to play “bottom” and earn 2 instead of following
the maximin strategy of playing “top”.
What is the Nash Equilibrium for the Prisoners’ Dilemma game
discussed above? For the two prisoners, the ideal outcome is one in which
neither confessed, so that they both get 6 months in prison. However,
confessing is a dominant strategy for each prisoner as it yields a higher
payoff regardless of the strategy of the other prisoner. Dominant strategies
are also maximin strategies. Therefore, the outcome in which both
prisoners confess is both a Nash Equilibrium and a maximin solution.
Thus in a very strong sense, it is rational for each prisoner to confess.
DYNAMICGAMES (REPEATED GAMES)
Two types of games played over time exist:
a) Games as above but repeated.
b) Sequential games in which one player moves before the other.
Finite Repeated Games
In most cases firms play games against each other repeatedly. For example a
price war game is played repeatedly. In a one shot game if one firm cuts
prices, the firm wins market share at the expense of the other firm that holds
prices without any negative consequences. However, in a repeated game
retaliation is possible.
Consider the following payoff matrix:
The way to solve finite repeated games is via backward induction: start with
the last period and work backwards.
Lets suppose the price war game is repeated 3 times. # Start with period 3.
This is equal to a one short game. The Nash Equilibrium is for both to cut
prices. Each firm need not fear retaliation in the next period because the
game is over. # Now look at PERIOD 2. Each firm knows that both firms
will cut prices in period 3. Period 3 is therefore irrelevant to period 2. There
is no hope that if you hold prices the other firm will do the same in the next
period, because you already know that the other firm will cut prices in
period 3. The Nash Equilibrium is again to cut prices for both firms in period
2. And so on. Thus the Nash Equilibrium in the repeated game is identical:
ALWAYS CUT PRICES. The payoffs are equal to 1 + 1 + 1 = 3 for both
players.
Infinitely Repeated Games
If the game never ends, the backward induction argument breaks down.
There is no final period in which the other firm will cut prices. The
mathematics of infinitely repeated games is too complex to discuss here, but
we can discuss the optimal strategy.
# A TIT FOR TAT strategy is to hold prices in the first period, and then in
the subsequent periods play the move your opponent played the period
before. Tit For Tat says that if your opponent cuts prices, you should
retaliate in the next period by cutting prices. If your opponent does not cut
prices, you should cooperate and hold prices in the next period. A famous
result is that Tit For Tat is a Nash Equilibrium in the infinitely repeated
game if firms care enough about the future.
Suppose you know your opponent will play Tit For Tat, then you have the
opportunity to get $5 million immediately by cutting prices. But the your
opponent will cut prices next period, and indeed every period thereafter
(forcing you to get $1 million each period), until you hold prices while your
opponent continues to cut prices. So you must get zero in order to persuade
your opponent to hold prices again. The sum of these two is 5 + 0 = 5 which
is less than 3 + 3 = 6 you get by simply continuing to hold prices knowing
that your opponent will cooperate as long as you do.
However if you do not care much about the future, it may be optimal to grab
the high profits today, and the hold prices later to persuade your opponent to
begin cooperating again. In general, Tit For Tat predicts that we will observe
long periods of mutual cooperation, or long periods of mutual retaliation.
The expected gains from cooperation outweigh those from undercutting.
Thus in a repeated game the PRISONERS” DILEMMA can have a
cooperative outcome.
SEQUENTIAL GAMES
In most of the games we have discussed so far, both players move at the
same time. There are many games where players do not move at the same
time for example:
 An advertising decision by one firm and the response by its
competitor,
 Entry deterring investment by an incumbent firm and the decision
whether to enter the market or not by a potential competitor, and
 A new government regulatory policy and the investment and output
response of the firms being regulated.
As an example let us return to the product choice problem discussed in the
previous section. This involves two companies that face a market in which
two new variations of breakfast cereal can be successfully introduced, as
long as each firm introduces only one variation. This time let us change the
payoff matrix slightly. As the above table shows, the sweet cereal will
inevitably be a better seller than the new crispy cereal, earning a profit of 20
rather than 10 (perhaps because consumers prefer sweet things to crispy
things). Both cereals will still be profitable as long as one firm introduces
only one cereal.
Suppose that both firms in ignorance of each other’s intentions announce
their decisions independently and simultaneously. Both will the probably
introduce the sweet cereal-and both will lose money.
Now suppose that Firm 1 can introduce its new cereal first (perhaps it can
gear up its production faster). We now have a sequential game: Firm 1
introduces a new cereal, and then Firm 2 introduces the other one. What will
be the outcome for this game? When making its decision, Firm 1 must
consider the rational response of its competitor. It knows that whichever
cereal it introduces, Firm 2 will introduce the other kind. Hence it will
introduce the sweet cereal, knowing that Firm 2 will respond by introducing
the crispy one.
This outcome can be deduced from the payoff matrix above but sequential
games are easier to visualize if we represent the possible moves in the form
of a decision tree. This is what is called the extensive form of the game. The
figure shows possible choices of Firm 1 (introduce a crispy or a sweet
cereal), and then the possible responses of Firm 2 for each of those choices.
The resulting payoffs are given at the end of each branch. For example if
Firm 1 produces a crispy cereal and Firm 2 responds by also producing a
crispy cereal, each firm will have a payoff of –5.
To find the solution to the extensive form game WORK BACKWARDS
from the end. For Firm 1, the best sequence of moves is the one is the one in
which it earns 20 and Firm 2 earns 10. Thus it should deduce that it should
produce the sweet cereal because then Firm 2’s best response is to produce
the crispy cereal. The advantage of moving first is that you can produce a
larger output as compared to the second mover.
THREATS, COMMITMENTS AND CREDIBILITY
In this section we look at the advantage a firm can have by moving first. We
will focus on the following question: What actions ca a Firm take to gain
advantage in the market place?
An action that gives a firm this kind of advantage is called a strategic move.
Thomas Schelling who first explained the concept and its implications gave
a good definition of a strategic move. A strategic move is one that
influences the other person’s choice in a manner favourable to oneself by
affecting other people’s expectations of how one will behave. One
constraints the partner’s choice by constraining one’s own behaviour. The
idea of constraining your own behaviour to gain advantage may seem
paradoxical but we will see that it is not. Lets consider the following
example:
Let us return once more to the product choice problem. The firm that
introduces its new breakfast cereal first will do best. But which firm will
introduce its cereal first? Even if both firms require the same amount of time
to gear up production, each has an incentive to commit itself first to the
sweet cereal. The key word is commit. If Firm 1 simply announces that it
will produce the sweet cereal, Firm 2 will have little reason to believe it.
After all Firm 2, knowing the incentives can make the announcement louder
and more vociferously. Firm 1 must constrain its own behaviour. In other
words Firm 2 must be convinced that Firm 1 has no choice but to produce
the sweet cereal. Such an action by Firm 1 might include an expensive
advertising campaign describing the new sweet cereal well before its
introduction, thereby putting Firm 1’s reputation on the line. Firm 1 might
also sign a contract for the forward delivery of a large quantity of sugar (and
make the contract public, or at least send a copy to Firm 2). The idea is for
Firm 1 to commit itself to produce the sweet cereal. Commitment is a
strategic move that will induce Firm 2 to make the decision Firm 1 wants it
to make, that is, to produce the crispy cereal.
Why can’t Firm simply threaten Firm 2 vowing to produce the sweet cereal
even if Firm 2 does the same? Because Firm 2 has little reason to believe the
threat and can make the same threat itself. A threat is only useful if it is
credible.
EMPTY THREATS
Suppose Firm 1 produces personal computers that can be used both as word
processors and to do other tasks. Firm produces only dedicated word
processors. Firm 2 produces only dedicated word processors. As the payoff
matrix in the table above shows, as long as Firm 1 charges a high price for
its computers both firms can make a good deal of money.
Even if Firm 2 charges a low price for its word processors, many people will
still buy Firm 1’s computers because they can do many other things although
some will be induced by the priced differential to buy the dedicated word
processor instead. However, if Firm 1 charges a low price for its computers
Firm 2 will also have to charge a low price or else make zero profit, and the
profit for both firms will be significantly reduced.
Firm 1 would prefer the outcome in the upper left hand corner of the matrix.
For Firm 2, however, charging a low price is clearly a dominant strategy.
Thus, the outcome in the upper right hand corner will prevail no matter
which firm sets its price first.
Firm 1 would probably be viewed as the DOMINANT firm in this industry
because its pricing actions will have the greatest impact on overall industry
profits. Then can’t Firm 1 induce Firm 2 to charge a high price by
threatening to charge a low price if Firm 2 charges a low price? No, as the
payoff matrix in the table above makes clear. Whatever Firm 2 does, Firm
will be much worse off if it charges a low price. As a result its threat is not
credible.
QUESTION 1
a) Explain what you understand by Prisoners’ Dilemma in Game theory.
b) Show and explain how come up with a solution in such a game.
c) Show and explain how you come up with a collusive equilibrium in
such a game.
d) Explain how you come up with a solution in an infinitely repeated
Prisoners’ Dilemma game.
QUESTION 2
Two firms are in the chocolate market. Each can choose to go for the high
end of the market (high quality) or the low end of the market (low quality).
The following payoff matrix gives the resulting profits.
a)
b)
c)
d)
What outcomes if any are Nash Equilibria?
What is the cooperative outcome?
If the manager of each firm is conservative and each follows a
maximin strategy, what will be the outcome?
Which firm benefits the most from the cooperative outcome? How
much would that firm need to offer the other to persuade it to
collude?
QUESTION 3
We can think of U.S. and Japanese trade policies as a Prisoners’ Dilemma.
The two countries are considering policies to open or close their import
markets. Suppose the payoff matrix is:
a) Assume that each country knows the payoff matrix and believes that
the other country will act in its own interest. Does either country have
a dominant strategy? What will be the equilibrium policies if each
country acts rationally to maximise its welfare?
b) Now assume that Japan is not certain U.S. will behave rationally. In
particular Japan is concerned that U.S. politicians may want to
personalize Japan even if that does not maximise U.S. welfare. How
might this affect Japan’s choice of strategy? How might this change
the equilibrium?
QUESTION 4
4. Two firms, A and B are thinking of entering a new market. The
payoff matrix for the two firms is given below:
The table shows profits in millions of dollars
a)
Calculate all (if any) Nash Equilibria.
b)
Calculate all (if any) dominant strategies by each firm.
c)
Rank all Nash Equilibria according to profit for firm A.
Is there an advantage to firm A moving first?
d)
Explain one problem that can occur for the two firms in this type
of game.