21. Game Theory Models Times Three
Deductive reasoning travels from the most abstract to the least abstract. It
begins with a set of axioms and uses the laws of logic and mathematics to
manipulate them to form predictions about the world.
—Rachel Croson
Game theory models strategic interactions. Many of the models that follow,
including our models of cooperation, signaling, mechanisms, and collection
action, involve games. We do not take up the analysis of games in much
depth because entire books are devoted to the subject. Our goal will be to
provide a gentle introduction. To that end, we present examples of the three
main classes of games: normal-form games, in which players choose from a
discrete set of actions (typically two); sequential games, in which players
choose actions sequentially; and continuous-action games, in which players
can choose actions of any magnitude or effect size. These examples
introduce the main concepts, help us to understand later models, and add
value in their own right.
The remainder of the chapter has four parts. We begin by covering 2-by2 zero-sum games. In a zero-sum game, each of two players chooses among
two actions. No matter what actions the players choose, the amount won by
one player is exactly offset by the losses of the other. We use zero-sum
games to define the basic terminology of game theory, to distinguish
between strategies and actions, and to introduce the concept of iterated
elimination of dominated strategies. We then study the Market Entry Game,
a sequential game, in which an entrant competes against an existing firm,
and we replicate that game many times to create what is known as the chain
store paradox. In the third part, we consider an effort game in which
individuals choose effort levels to win a prize of a fixed amount. Increasing
effort improves a player’s chances of winning the prize. The chapter
concludes with a brief discussion of the value of game theory models
generally.
Normal-Form Zero-Sum Games
In this section we analyze two-player normal-form zero-sum games. In both
games, each player chooses an action and receives a payoff that depends on
the player’s own action and the other player’s action. In addition, the
players’ payoffs sum to zero. In the first game, Matching Pennies shown in
figure 21.1, each player chooses one of two actions: heads or tails. The row
player wants to match the other player’s choice, and the column player
wants to mismatch. Payoffs are shown in the matrix below:
Figure 21.1: The Matching Pennies Game
A strategy is a rule for how to play the game. It could be a choice of a
single action, a randomization between actions, or, as we see in the next
section, a sequence of actions. A Nash equilibrium of a game is a pair of
strategies such that each player’s strategy is optimal given the strategy of
the other player. In Matching Pennies, in the unique equilibrium strategy
both players randomize equally between the two actions. To prove
randomization is an equilibrium, we need to show that if each player
randomizes, the other player cannot do better than randomizing. This is
straightforward. If the row player (actions and payoffs in bold) plays heads
with probability and tails with probability , then the column player earns
zero regardless of her action. Therefore, randomizing is also an optimal
strategy for the column player. By symmetry, randomization is optimal for
the row player as well.
This optimality of randomization has implications for behavior in
strategic settings. Sports are zero-sum: one team (or player) wins and one
loses. During penalty kicks, a striker wants to randomize between aiming
for the left or right corner. In tennis, a server wants to randomize serving to
the inside or outside. On fourth and goal in football, an offense wants to
randomize between run and pass. In each case, the opponent also wants to
randomize their planned responses. Any non-randomness can be exploited.
The same holds in card games such as poker. A good poker player bluffs
randomly. If she always bluffed, her opponents would learn her strategy and
stay in the game. She would then lose every time she bluffed. Similarly, if
she never bluffed, her opponents would learn to fold. Optimal bluffing
makes her opponents uncertain whether to stay or fold.
In our second game, the Minimize Risk Game shown in figure 21.2 each
player can take a risky action or a safe action. This is an asymmetric zerosum game. The payoffs depend not just on the actions but also on which
player takes which action. In this game, the row player has a dominant
strategy to play safe. No matter what action the column player chooses, the
row player earns a higher payoff by choosing safe. The column player does
not have a dominant strategy. If the row player chooses risky, the column
player should choose risky. If the row player chooses safe, the column
player should choose safe.
Figure 21.2: The Minimize Risk Game
By thinking through the incentives for the row player, the column player
can deduce that the row player will never choose risky because risky is
dominated by safe. Therefore, the column player knows that the row player
will choose safe. Given that, the column player should also choose safe.
This type of reasoning in which one player rules out dominated strategies
for the other player is known as iterative elimination of dominated
strategies. In this game, using iterative elimination of dominated strategies
shows that both players choosing safe is the unique Nash equilibrium.
Sequential Games
In a sequential game, players take actions in a specific order, as shown on a
game tree, which consists of nodes and edges. Each node corresponds to a
moment when a player must take an action. Each edge from that node
denotes one of the possible actions. At the end branches of a game tree, we
write the payoffs associated with following that path of actions. The game
tree in figure 21.3 shows the Market Entry Game.
Figure 21.3: The Market Entry Game
The Market Entry Game involves two players: a potential entrant and an
existing firm. If the entrant chooses not to enter the market (the left branch
of the tree), it earns no payoff and the existing firm earns a profit of 5. If the
entrant enters the market, the existing firm must choose between accepting
the new entrant and seeing its profits fall from 5 to 2 or competing with the
new firm and driving its profits to zero and the entrant’s profits negative.
We assume the entrant’s profits to be negative because it has to pay for the
cost of entering.
In a sequential game, a strategy corresponds to an action choice at each
node. Suppose that the existing firm chooses to compete if the entrant
enters. If the entrant knows this, the entrant would not enter, as doing so
would produce negative profits. This set of actions, the entrant choosing to
not enter and the existing firm planning to compete if the entrant did enter,
are a Nash equilibrium. However, this is not the only Nash equilibrium, nor
is it the most likely outcome. There is a second equilibrium in which the
entrant chooses to enter the market and the existing firm accepts the
entrants move and does not compete.
To select among these two equilibria, we apply a refinement criterion. In
sequential games, a common refinement chooses the subgame perfect
equilibrium. We solve for the subgame equilibrium using backward
induction: we start at the end nodes and choose the optimal action at each.
We then work backward up the game tree assuming that each player
chooses the best action given the actions of the other player at subsequent
nodes. In the Market Entry Game, we start at the end node for the existing
firm. It has an optimal action: to accept. We then move up the game tree
and see that the entrant has an optimal strategy to enter.
This game becomes even more interesting when replicated. Imagine that
the firm exists in many markets. Perhaps it is a chain store with franchises
in dozens of cities. Suppose also that there exists a sequence of potential
entrants. The firm is going to play one Market Entry Game after another in
sequence.
If the firm reasons using backward induction starting from the last
market, it will accept the entrant in that last market, as that is the payoff
maximizing action. Continuing with the same logic, the firm will accept the
second-to-last entrant. It will also accept every other entrant. It follows that
in the unique subgame perfect equilibrium of the sequence of games, all of
the potential entrants choose to enter, and the firm accepts all of them.
Though entrance and acceptance in every market is the unique subgame
perfect equilibrium. In practice, it may not occur. Imagine we are on the
board of directors of the existing firm and we are confronting the first
entrant, who (having studied game theory) enters the market. We may want
to compete to try to deter entry in the other markets. Competing would be
an intelligent strategy if it is credible, that is, if we can build a reputation as
willing to compete. The outcome we hope to create differs from the
subgame perfect equilibrium.
Game theorists refer to the disconnect in this game between what game
theory predicts and what actual players would try to produce as the chain
store paradox. It is one example where what game theory considers to be
optimal behavior may not be the behavior chosen by a sophisticated player
when the stakes are large. The example does not disprove game theory or
undermine the rational choice assumption, so much as it reveals why we
must always challenge assumptions.
Continuous Action Games
We now study a game in which players choose from a continuum of
possible actions. In the game, actions correspond to effort levels. By
choosing greater effort, a player increases her probability of winning a
prize. This game allows us to model any number of players.
The derived expression for equilibrium effort reveals a number of
insights. As we would expect, individual effort increases with the size of the
prize. Also, in equilibrium, total effort will be less than the value of the
prize. That too would be expected given that we assume players optimize.
Players should put forth effort to win, but not an unreasonable amount.
The Effort Game
Each of N players chooses an effort level expressible in monetary
terms to win a prize of value M. The probability that a player wins the
prize equals her effort divided by the total effort of all players. If Ei
equals the effort level of player i, her probability of winning is given
by the following equation:1
Equilibrium:
We can see the effects on individual and total effort by increasing the
number of players. Here, the findings are less intuitive. According to the
model, individual players’ effort levels decrease but the total effort by all
players increases. Thus, the model implies that efforts by organizers of
research grant opportunities, architectural competitions, and essay contests
to attract large numbers of entrants may, paradoxically, produce lowerquality winners because in the larger contests, participants have less
incentive to put in effort.
Summary
We began this chapter by covering zero-sum games. These games assume
no mutually beneficial combinations of actions. Any action that proves
good for one player necessarily hurts another player. In a zero-sum
decision, any action harms someone as much as it helps someone else.
Taking money from one person and giving it to another is a zero-sum
action. Many personal actions and policy choices are zero-sum in at least
one dimension. We have only so many hours in the day, so much money to
spend, and so many resources to allocate. That said, a zero-sum action in
one dimension may not be zero-sum in another. A budget relocation could
be zero-sum in monetary terms but positive or negative sum in terms of
human happiness or fulfillment.
We should always explore whether a proposed policy change creates a
zero sum game. For example, many people argue for school choice—giving
parents the ability to choose the school their child will attend—because it
increases competition. Market logic suggests that by being forced to
compete, schools have incentives to improve quality.
However, schools only have an incentive to improve quality if excess
capacity exists. Otherwise, school choice can create a zero-sum game
among the students. Imagine a city with 10,000 students and 10 schools
each with a capacity for 1,000 students. If the students rank the schools in
the same order, spots in the best schools will have to be allocated by lottery.
Those who win the lottery will go to better schools. Those who lose the
lottery will go to worse schools, who remain in operation because of a lack
of excess supply. Students are playing a zero-sum game. If new schools
open, or if existing schools improve, the game is no longer zero-sum.
Everyone can win.
Both the market and the zero-sum models provide insights. The market
model reveals incentives for quality improvements and for the creation of
new schools. The zero-sum model shows that school choice alone means
that some students will gain while others lose. The relative weight we
should place on each model depends on the context: Does sufficient excess
capacity exist in the better schools so that they can absorb the additional
students? Do schools have the resources and expertise to improve their
quality? Will entrepreneurs create new schools? Does the transportation
system enable students to get to multiple schools in order to create
competition?
Our takeaway should be that neither of the two models gives us a correct
answer, but each produces useful insights. School choice will create
competition. It also creates a massive sorting problem with features of a
zero-sum game. Whether the positive aspects of competition will outweigh
the negative sorting costs depends on the context. We must array our lattice
of models on the set of facts to make a good policy choice.
The Identification Problem
Data on people’s actions often reveals clustered behavior. Good
students are more likely to be friends with other good students than
with students who struggle. People who engage in criminal behavior
are more likely to interact with other people who commit crimes than
are people who do not commit crimes. Any number of social goods
and ills—smoking, physical fitness, obesity, and even happiness—
cluster in social networks. People also cluster by beliefs: Democrats
cluster. Republicans cluster. Libertarians cluster.
We have two models that can explain clustering: peer effect models
and sorting models. Peer effect models explain clustering with game
theory. Individuals play a coordination game with their friends. In
sorting models, people move to be near others who are like them. A
cluster of good students could result from either students coordinating
on a common behavior (peer effects) or arise because good students
choose to hang out with one another (sorting). Given a snapshot of
data, the two are indistinguishable.
Data: Students earn either high H or moderate M scores, with each
being equally likely. Students belong to friendship cliques of size 4
with the following distribution: p({H, H, H, H}) = P({M, M, M, M}) =
, p({H, H, H, M}) = P({M, M, M, H}) = , and p({H, H, M, M}) =
0.
Peer effect model: Students originally form random groups of size 4:
p({H, H, H, H}) = P({M, M, M, M}) = , p({H, H, H, M}) = P({M,
M, M, H}) = , and p({H, H, M, M}) = People who belong to groups
consisting of only one type remain unchanged. A person who has the
opposite type from everyone else switches type, so an {H, H, H, M}
group becomes {H, H, H, H}. In groups with equal numbers of each
type, one member switches type. The group {H, H, M, M} is equally
likely to become {H, H, H, M} or {M, M, M, H}.
Sorting model: Students originally form random groups of size 4. In
any group with two types, a person who has the opposite type as at
least two other people switches groups with someone of the other
type. It follows that {H, H, H, M} becomes {H, H, H, H} and {M, M,
M, H} becomes {M, M, M, M}, and that any group of the form {H, H,
M, M} is equally likely to become {H, H, H, M} or {M, M, M, H}.
Both models are consistent with the data, creating an
identification problem. With only a snapshot of data, we cannot
determine whether smoking, reading manga, or longboard
skateboarding are peer effects or sorting. In some instances, we can
reason through which model applies. The tendency for people to say
“pop” in the Midwest and “soda” on the coasts is something that we
can safely assume to be driven by peer effects—few people move to
Boston so that they can refer to Coke as “soda.” On more important
behaviors such as educational performance, drug use, obesity, and
happiness, we need time series data to discern which model applies.
By looking across time, we can discern if people change their
behaviors to fit in with their friends (peer effect), or if they change
their friends and retain their behaviors (sorting). In many cases of
interest such as school performance, both effects may be in play.2