5.3 Classic Game Models
[5.2
Using Game Theory]
[5.4 Simultaneous Games]
[5.5 Sequential Games]
[5.6 Oligopoly]
[5.7 Network Effects]
In the field of game theory, several games are used routinely to
illustrate various concepts. These games are so widely used within this field
of study that you will need to familiarise yourself with them. This topic
describes these classic games and the situations they represent.
The following are commonly used game models:
·
The Prisoner’s Dilemma
·
The Battle of the Sexes
·
The Deer Hunt
·
The Lock Out Game
·
The Chicken Game
Background
Two
people have been arrested on the suspicion of having robbed a bank. No stolen
money has been recovered from them, and the evidence against them is purely
circumstantial. There is sufficient evidence, however, to convict each of a
minor weapons violation. The prisoners are kept in separate cells and are not
permitted to communicate. They are then asked to simultaneously choose one of
two possible actions — confess to the robbery or deny involvement in the
robbery. (In game theory, these two options are usually referred to as confess
and not confess).
The prisoners are told that if they confess, the weapons charge will be
dropped. They are also told that if one prisoner confesses and the other does
not, then the confessing prisoner will be set free, because he co-operated, and
the non-confessing
prisoner will be convicted of both the weapons charge and the robbery. On the
other hand, if they both confess, they will both be convicted of the robbery.
Co-operation
can occur in the Prisoner's Dilemma if there is no known end period. Click on
the link here to see how.
Note that the payoff numbers provided in the examples below are
arbitrary and are only intended to illustrate the concept, not any actual
result.
Simultaneous version
In this classic game, two prisoners are
choosing between confessing (C) and not confessing (NC) to the bank robbery
simultaneously. Because they are moving simultaneously, each prisoner must
choose a strategy without any information about the strategy chosen by the
other prisoner. The payoffs corresponding to the different strategy
combinations appear in the following payoff matrix, which is discussed
further in topic 5.4:
|
|
Prisoner 2
|
|
C
|
NC
|
|
Prisoner 1
|
C
|
10, 10
|
150, 0
|
|
NC
|
0, 150
|
100, 100
|
Both prisoners will be convicted for the minor weapons violation if
neither prisoner confesses. But after their prison term is over, they will be
able to use the money that they stole and hid. Therefore, the strategy
combination where both prisoners choose Not Confess gives them a payoff of 100
each. However, if both choose Confess they will both be convicted of the
robbery. This strategy combination gives them a low payoff of 10 each.
If either prisoner chooses Not Confess and the other chooses Confess,
the prisoner who confesses gets a payoff of 150 (which represents freedom and
also the stolen money) while the prisoner who chose not to confess gets the
lowest payoff of 0.
The main feature of the Prisoner's Dilemma game is that each prisoner
has a dominant strategy, which is to Confess.
This is a strategy that offers the largest payoff regardless of the strategy
chosen by the other prisoner.
Consider the choices of Prisoner 1 in the
payoff matrix above. If Prisoner 2 chooses Confess, then Prisoner 1 can get
payoffs of 10 from Confess and 0 from Not Confess. Clearly, Confess is the
better strategy if Prisoner 2 chooses Confess. Similarly, if Prisoner 2 chooses
Not Confess, then Prisoner 1 can get payoffs of 150 from Confess and 100 from
Not Confess. Again, Confess is the better strategy if Prisoner 2 chooses Not
Confess.
We can arrive at the equilibrium of the Prisoner's Dilemma game by
noting that a rational player will always play a dominant strategy (Confess)
and that it is not rational to choose a dominated strategy (Not Confess).
Therefore, the strategy combination (Confess, Confess) is a dominant strategy equilibrium of this
game.
Click on the link here to see
how repeating the simultaneous
version of Prisoner's Dilemma can impact the equilibrium.
The fact that each prisoner chooses Confess is a surprising outcome. By
choosing their dominant strategy, each prisoner gets a payoff of only 10.
However, they could have done much better and earned a payoff of 100 if they
had each chosen the dominated strategy Not Confess. This is one of the most
interesting aspects of the Prisoner's Dilemma game: it is possible for each
person to follow a strategy that is rational from his or her individual point
of view even though a different choice would have yielded a better outcome.
The Prisoner's Dilemma game is also important because it arises in many
real-world situations. It can explain the decision of nuclear-armed political
adversaries (such as India and Pakistan or the United States and the former
Soviet Union) to build nuclear weapons even though it is in their collective
interest to dismantle such weapons. (To translate the game to a nuclear
standoff, replace "Confess" with "Have Nuclear Weapons" and
"Not Confess" with "Destroy Nuclear Weapons." In this
version of the game, it becomes a strictly dominant strategy for each country
to Have Nuclear Weapons).
The Prisoner's Dilemma can also explain why many firms (for instance, in
the airline industry) are locked into charging low prices even though the
market could support higher prices. To play the game in terms of airline
pricing, replace "Confess" with "Charge Low Prices” and
"Not Confess" with "Charge High Prices”. As in the Prisoner's
Dilemma, charging low prices is a strictly dominant strategy for each firm,
even though the firms would be collectively better off by charging a higher
price. This would require, of course, that they collude, which is illegal for
competing firms to do in most countries.
Sequential version
Suppose that the prisoners make their
moves sequentially instead of simultaneously. In the sequential game, prisoner
1 moves first and chooses Confess or Not Confess. Prisoner 2 observes the
action of prisoner 1 and then chooses Confess or Not Confess. The difference
lies in the fact that one prisoner can observe the action of the other. Of
course, the two prisoners are not allowed to make enforceable commitments to
choose one action over another. The game tree discussed in further
detail in topic 5.5, would be:
(Don’t worry if you can’t fully understand the payoff matrix yet. It
will be dealt with in more detail in Topic 5.5.)
Will allowing the prisoners to move sequentially change the outcome of
the game? You can use backward induction to see.
To do so, start in the second stage of the game after Prisoner 1 has
made a choice. If Prisoner 1 has chosen Confess, then the game is at node b. At this node, Prisoner 2's optimal
decision is also to choose Confess. If Prisoner 1 has chosen Not Confess, then
the game is at node c and once again
Prisoner 2's optimal decision is to choose Confess. We can now fold the game
backwards to the first stage where Prisoner 1 faces the following choice:
Prisoner 1's optimal choice, given that Prisoner 2 will always choose
Confess in the second stage, is to choose Confess as well. Therefore, the
outcome is the same whether the game is played simultaneously or sequentially.
Click on the link here to read an example
that parallels the Prisoner's Dilemma classic game:
Background
A husband and wife are
deciding whether to spend the evening at a Bullfight or the Opera. The husband
prefers the bullfight to the opera, but the wife prefers the opera to the
bullfight. Both would prefer to be together at either event rather than
spending the evening apart. The worst outcome would be for the husband to spend
the evening alone at the opera and the wife to spend the evening alone at the
bullfight. (Note that the situation of this game is a little antiquated.
However, it does capture issues where individuals wish to co-ordinate their
actions but face a dispute as to what actions they should choose).
Note that the payoff numbers provided in the examples below are
arbitrary and only intended to illustrate the concept, not any actual result.
Simultaneous version
The husband and wife are choosing
simultaneously between Opera (O) or Bullfight (BF). Their preferences are
reflected in the following payoff matrix:
|
|
Husband
|
|
O
|
BF
|
|
Wife
|
O
|
5, 4
|
3, 3
|
|
BF
|
0, 0
|
4, 5
|
When played simultaneously, the Battle of the Sexes has two Nash equilibria:
both choose Bullfight or both choose Opera. This is because if one spouse
chooses Bullfight or Opera, then it is optimal for the other to choose the same
because the couple would prefer to spend the evening together. The Battle of
the Sexes draws attention to the problem of co-ordination in games in which
players have divergent preferences but a common interest in co-ordinating their
strategies.
An economic example corresponding to this game would be two firms
competing over the adoption of an industry-wide standard for a product they
have developed. The two firms would obviously have different preferences over
the standard they would like the industry to adopt; however, both would like to
co-ordinate on the same standard because consumers are more likely to buy a
product with a common standard across the industry.
The Battle of the Sexes game stresses the need for a co-ordinating
mechanism. This can point the players towards a particular Nash equilibrium
among the many that exist.
Sequential version
Consider the sequential version of the
Battle of the Sexes in which the wife moves first, and the husband moves second
after observing the action of the wife. The game tree model for this game would be:
We can solve this game by backward induction. In the second stage, if the
game is at node b, the optimal choice
of the husband is Opera; while if the game is at node c, the optimal choice of the husband is Bullfight. Folding the game
to the first stage, the wife faces the following payoffs:
The optimal choice of the wife is therefore to choose Opera. In the
backward induction outcome of the sequential game, both players choose Opera.
The sequential version of the Battle of the Sexes highlights an
important principle: First-mover advantage. The wife knows that the husband is rational and will always
choose a best response to any action she chooses in the first stage. Therefore,
she chooses Opera knowing that the husband's best response will be to choose
Opera as well.
Another co-ordinating mechanism is pre-play communication. If the
husband and wife can discuss their strategies prior to deciding, they may argue
a great deal, but they are likely to end up co-ordinating their choices.
Click on the link here to read an example
that parallels the Battle of the Sexes classic game.
Background
Two wolves are hunting for deer in the forest. If they work together,
they can get a deer, which will provide a great deal of meat for both. However,
if one wolf goes off to hunt rabbit, that wolf will get at least a few rabbits,
whereas the wolf that continues to hunt deer will get nothing. Alternatively,
both wolves can choose to hunt rabbit only. However, since they are both
hunting rabbit, each will interfere with the other's ability to effectively
hunt rabbit.
Note that the payoff numbers provided in the examples below are
arbitrary and only intended to illustrate the concept, not any actual result.
Simultaneous version
In this game, two hunters must choose
between hunting deer or hunting rabbit. The payoffs are shown in the payoff matrix below:
|
|
Wolf 2
|
|
Deer
|
Rabbit
|
|
Wolf 1
|
Deer
|
10, 10
|
0, 8
|
|
Rabbit
|
8, 0
|
6, 6
|
This game has two Nash equilibria: one
in which both wolves choose Deer and the other in which both choose Rabbit.
The Deer Hunt game highlights the problem of getting players to
co-operate on the more rewarding activity of hunting deer (since each wolf gets
a higher payoff of 10) instead of the less rewarding activity of hunting rabbit
(in which each wolf gets a payoff of only 6). It may seem that if the two
wolves are allowed to communicate before playing the game, then they may agree
that it is in their interest to hunt deer because it yields higher payoffs for
both relative to hunting rabbit. To summarise, hunting deer will be the focal
point of the game.
However, many game theorists have argued that the wolves might still end
up choosing to hunt rabbit. This is because if one wolf believes that the other
wolf can make an error and choose the wrong strategy, then hunting deer is
risky. If wolf 1 chooses Deer, and wolf 2 makes an error and chooses Rabbit,
then wolf 1 will suffer a payoff of 0.
Hunting rabbit is not as risky because it does not inflict the same
magnitude of losses in the event of an error. Moreover, both wolves know that
the other is making the same assessments so each might rationally expect the
other to choose to hunt rabbit. Therefore, the Nash equilibrium where both
wolves hunt rabbit often occurs.
Sequential version
Suppose that the wolves are choosing
between hunting rabbit and deer sequentially. The sequential version game tree is shown here:
Is it possible to induce the wolves to co-operate and always choose Deer
in a sequential version of the game?
To arrive at the backward induction solution, start in the
second stage. At node b, the optimal
decision of wolf 2 is Deer, while at node c
the optimal decision is Rabbit. Now, fold the game to the first stage:
The optimal decision for wolf 1 is to choose Deer. Therefore, both
wolves choose Deer in the backward induction solution to the game.
There is a sharp contrast to the simultaneous version of the game where
the wolves may co-ordinate on the less risky Nash equilibrium in which both
hunt rabbit. With simultaneous games, there is always an element of risk
because one wolf cannot observe the strategy of the other and may therefore
choose an incorrect move. In the sequential game, there is no risk because wolf
1 moves first and commits to a strategy. Wolf 2 can observe wolf 1's strategy
and choose the best response to it. Therefore, Wolf 1 can go ahead and choose
to hunt deer knowing that wolf 2's best response will also be to hunt deer.
Click on the link here to read an example
that parallels the classic Deer Hunt game.
We will also investigate an application of
this type of game when we examine network effects in Topic 5.7.
Background
A monopolist
in a certain market (the
incumbent) is threatened by the potential entry of another firm (the entrant).
The entrant has the option of entering or not entering the market. The
incumbent has the option of fighting the entry (if it occurs) by dropping its
prices substantially or simply accepting the entry by lowering its prices to a
lesser extent. The incumbent would rather that the entrant chooses not to enter
the market so that it can maintain its monopoly. If the entrant is allowed into
the market, the incumbent's profits will drop substantially. Obviously, if it
chooses to fight the entry after it occurs, the incumbent's profits will drop
even more.
The incumbent knows that entry is profitable to the entrant only if the
incumbent accepts (does not fight) entry. Therefore, the incumbent has an
incentive to announce that it will institute large discounts if there is entry
into the market. It is extremely important that both players have full
knowledge of the structure of the game, including all payoffs. If this
assumption is violated, an incumbent might fight to indicate to other firms
that it is in a very strong position — whether that is true or not — when in reality
not fighting might be more profitable than fighting a firm that has already
entered the market.
Please note that the Lock Out Game described in this subject is a
two-stage version of the classic Chain Store Game, which typically takes place
over many stages. Additionally, please note that the payoff numbers provided in
the examples below are arbitrary and only intended to illustrate the concept,
not any actual result.
Simultaneous version
Suppose
that the two firms are choosing their strategy simultaneously. The entrant
chooses between entering the market (E) and not entering the market (NE). The
incumbent chooses between fighting the entry (F) and accepting the entry (A).
The payoffs appear in the following payoff matrix:
|
|
Incumbent
|
|
F
|
A
|
|
Entrant
|
E
|
-2, 0
|
3, 2
|
|
NE
|
0, 5
|
0, 5
|
In this case,
there are two Nash equilibria:
one in which the incumbent chooses to fight entry (F)
and the entrant chooses not to enter (NE), and one in which the incumbent
chooses to accept entry (A) and the entrant chooses to enter (E). It is obvious
that the incumbent would prefer the Nash equilibrium in which the potential entrant
does not enter. However, as we will see, in the more realistic case where moves
are made sequentially, this Nash equilibrium is more than likely unobtainable.
Sequential version
Suppose
that the two firms — the entrant and the incumbent — are moving sequentially.
Specifically, suppose that the entrant chooses between entering the market (E)
and not entering the market (NE). Then the incumbent decides between fighting
the entry (F) or accepting the entry (A). The game tree model would be as follows:
Suppose the
incumbent threatens to fight the entrant if it chooses to enter. What is the
entrant's best strategy at that point?
Using backward induction to solve this problem, you
will notice that once the entrant has committed to enter the market, the best
thing for the incumbent to do at that point is to accept the entry. Therefore,
the entrant should disregard the incumbent's threat and proceed to enter the
market since entry would give a higher payoff.
This
demonstrates the importance of commitment in a game. If the incumbent could
credibly commit to fighting, then the entrant would rationally choose not to
enter. However, since the incumbent cannot do so, the entrant will choose to
enter.
Does the fighting ever really stop?
It is
interesting to note that when an incumbent adopts a strategy that allows an
entrant to initially come into a market without a fight, it is different to the
incumbent surrendering entirely. It is a tactical move designed to make the
best use of the incumbent's resources. In the real world, both the incumbent
and entrant will fight for market share and customer loyalty on an ongoing
basis.
Softsoap Liquid Soap
Background
Two
people are driving their cars towards each other at high speed. As their cars
come closer, each driver has two choices — to go straight ahead (to act
bravely) or to swerve to avoid the other car (to act cowardly or like a
chicken). If both players choose to go straight ahead, then they will collide
head on with disastrous consequences. If both swerve, then a crash is avoided
but both drivers are shown to be "chicken". However, if one chooses
to go straight ahead and the other chooses to swerve, then the nonswerving player is shown to be "brave" while
his opponent is shown to be "chicken".
Note that
the payoff numbers provided in the examples below are arbitrary and only
intended to illustrate the concept, not any actual result.
Simultaneous version
Both drivers must simultaneously decide between
swerving and going straight ahead. The payoffs for each player are shown in the
following payoff matrix:
|
|
Player 2
|
|
Go
|
Swerve
|
|
Player 1
|
Go
|
0, 0
|
5, 1
|
|
Swerve
|
1, 5
|
4, 4
|
If one player chooses Swerve, then the other will want to choose Go.
Also, if one player chooses Go, the other will naturally choose Swerve. Thus,
there are two Nash equilibria in
pure strategies of the Chicken Game. Both involve one player choosing Go and
the other choosing Swerve. What the outcome will actually be (eg, which player will end up doing what) depends on many
factors, including past actions and the perceived aggressiveness of opponents.
Two firms competing to enter a new industry that can support only one
firm (a natural monopoly) is a situation that parallels the Chicken Game. The
Chicken Game once again highlights the problem of how players co-ordinate on a
Nash equilibrium, particularly if some of the Nash equilibria
(such as the pure strategy Nash equilibria in this
case) provide inequitable payoffs to the players.
Sequential version
Suppose
the players make the decision to swerve or go straight sequentially. (Although
this makes less sense in this specific context, there are several games that
are similar to the Chicken Game in which sequential moves would be
sensible).The game tree model for this situation would
be as follows:
What is
the backward induction outcome of the
sequential version of the Chicken Game? Is there a first-mover advantage in
this game?
Starting
at the second stage, at node b, the
optimal decision for player 2 is to swerve while at node c, the optimal decision is to go. Folding the game to the first
stage then shows that the optimal decision of player 1 is to go. Therefore,
there is a first-mover advantage. The player moving first can choose to go
knowing that since the player moving second is rational, he or she will play
the best response and swerve.
Click on the link below to read an example that parallels the classic Chicken Game.