Jennifer Bachand
Math 482H
Game Theory
Game theory is defined as the logical analysis of situations of conflict and
cooperation. A key question in game theory is to reason about the behavior we
should expect to see when players take part in a given game.
First, we will discuss what a mathematical game exactly is. A mathematical game
has the following properties:
• At least two players (by “players” we could mean anything, such as an
individual, company, or entity)
• Each player has a number of possible strategies
• The strategies chosen determine the outcome
• Each possible outcome has its associated “payoff”, representing the value of
the outcome to that player
Some examples of mathematical games are chess, bridge, and poker. Game theory
generalizes the study of these and applies it to the real world.
Note: Game theory models outcomes based on assumed rational play. This limits its
application as real-world games are enormously complex and humans do not
usually behave rationally.
An example of a model of a game and its payoff table:
Rose
A
B
C
A
Colin
(-2,2)
(0,0)
(-5,5)
B
(-3,3)
(-2,2)
Table 1
(10,-10)
where Rose and Colin are players with strategies A, B, and C.
This is an example of a zero-sum game, meaning that the payoffs for each
outcome add to 0. Zero-sum games represent situations of pure conflict, where one
player’s gain is directly the other player’s loss.
We can simplify the situation by looking at the game through Rose’s
perspective. We thus consider only her payoffs (Table 2) and then the motion
diagram (Table 3), which is where we draw an arrow to the smallest entry in each
row and the largest in each column.
A
A
B
0
2
2
B
C
-3
-5
A
C
Table 2. This is Table 1 but from
Rose’s perspective (Payoff Table)
10
A
B
0
2
2
B
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Math 482H
-3
-5
Table 3, a motion diagram for
Table 2.
10
From these tables, we see that we should play strategy C, with the hope that
Colin will play B. However, if Colin plays A, the consequences will be the worst
possible. But if Colin was planning to play strategy B, then Rose should play strategy
A; this continues.
We say that Strategy S dominates strategy T if every outcome in S is at least
as good as the corresponding outcome in T, and at least one outcome in S is strictly
better than an outcome in T. i.e., S ≥T for all entries.
Example:
Rose\Colin A
B
C
D
Rose/Colin A
C
2
4
3
C
A
12
D
-16
B
5
3
-1
1
0
1
7
0
0
-20
16
Table 4. An example payoff table for Rose
A
B
B
C
D
D
Table 5. Motion diagram for Table 4.
Here, we see that Colin B dominates Colin C because Rose gains less from
Colin B, no matter what she plays.
Jennifer Bachand
Math 482H
Dominance Principle: A rational player should never play a dominated strategy.
We may note that Rose C and Colin B are the most cautious strategies in this
game, because by playing them, Rose knows she will gain at least 2 and Colin knows
that he will lose no more than 2. Rose C/Colin B is called an equilibrium outcome,
because they could not have forced a better outcome. Is there a way to maximize the
occurrence of an equilibrium outcome? This brings us to the concept of equilibrium
strategy.
Consider an mxn game:
A\B
A1
A2
…
Am
βj
Let
B1
a11
a21
…
am1
β1
B2
a12
a22
…
am2
β2
…
…
…
…
…
…
Bn
a1n
a2n
…
amn
βn
αi
α1
α2
…
αm
(αi) = min(over j) aij= minimum of the numbers in the ith row.
This is the "most we can win"; our opponent will respond with strategy Bj for
minimum yield, αij
When choosing strategy A0 we must count on the fact that by a suitable
action, our opponent can keep us from winning more than αi. So we stop at strategy
Ai, where αi is a maximum.
α = max(over i) αi
Thus,
α = max(over i) min(over j) aij.
This is called the lower value of the game, the maximum of the minimum
yield, or the maximin. Its corresponding strategy is called the maximin strategy;
by playing the maximin strategy, we are assured a yield of greater than or equal to α
in any case.
Similarly, our opponent is trying to reduce our yield to the minimum, and
examines each of our strategies from the point of view of maximum yield for that
strategy.
We set
βi= max(over i) of aij.
We find the minimum of βj:
β= min(over j) βj.
Jennifer Bachand
Math 482H
So
β=min (over j) max(over i) aij.
β is called the upper value of the game, or minimax. Its corresponding
strategy is called the minimax strategy. By playing the minimax strategy, our
opponent ensures that he or she will not lose more than the amount β, regardless of
our adopted strategy. This is the Priniciple of Minimax.
We further explain with the following example:
A/B
A1
B1
2
A2
-3
βj
A3
B2
B3
-3
αi
4
-3
6
-5
4
4
-5
-5
4
4
6
Table 7
-5
The maximin strategy here is A1 (we lose no more than 3) and the minimax
strategy here is B1 or B2 (opponent loses no more than 4).
Sometimes, α=β. If α=β we call this value v, or the common value of the
game. This represents a saddle point, and the entry at this outcome is ≤ any in its
row and ≥ any in its column.
Saddle Point Principle: If a matrix has a saddle point, both players should play a
strategy which contains it. That is, if one party adheres to their optimal strategy and
the other deviates, the deviating party can only decrease their yield.
Saddle points are equivalent and interchangeable; any two saddle points in
a matrix game have the same value. If Rose and Colin both play strategies containing
a saddle point outcome, result will always be a saddle point. We confirm this with
the following proof:
Suppose a and b are saddle point entries in a matrix game, and c and d are
other entries at the corners of a rectangle containing a and b.
a
…
…
…
c
d
…
…
b
Jennifer Bachand
Math 482H
Since a is smallest in its row and b is largest in its column, a≤c≤b. Since b is
smallest in its row and a is largest in its column, b≤d≤a.
So a≤c≤b≤d≤a, so a=c=b=d.
So c and d are also the largest in their columns and smallest in their rows,
and thus are saddle points.
We will now go on to discuss mixed strategies. In most matrix games, the row
maximin is not equal to the column minimax, which means there is no saddle point.
Thus, the optimal strategy for each player, rather than one pure strategy, is a mixed
strategy: this is where each party plays a mixture of strategies according to a
certain fixed probability. We analyze this effect using expected value. The expected
value of payoffs a1, a2,..ak with respective probabilities p1, p2, …,pk is
E(x) = p1a1+p2a2+….+pkak,
or the weighted average of those payoffs. If a party knows its opponent is playing a
given mixed strategy and will continue to play it regardless of this party's play, the
expected value principle states that the party should play its strategy with the
largest expected value.
Here is an example:
Colin
Rose
A
B
βj
A
B
2
-3
2
3
0
3
αi
-3
Table 8
0
In this example, the maximin is 0, which does not equal the minimax, which is
2. If Colin flips a coin to play CA 50% of the time and CB 50% of the time, then our
expected payoffs are:
RoseA = .5(2)+ .5(-3)= -.5
RoseB = .5(0) + .5(3)= 1.5
Thus, Rose should play strategy B if she knows Colin's mixed strategy.
However, Colin understands that if he uses a ½ CA, ½ CB mixed strategy,
Rosie can take advantage of this. So instead, he will use CA with a probability of x
and CB with a probability of (1-x), 0<x<1.
Our new expected values become:
Jennifer Bachand
Math 482H
RoseA = x(2) + (1-x)(-3) = -3 + 5x
RoseB = x(0) + (1-x)(3) = 3-3x
If these are the same, Rose can't take advantage of Colin's strategy. So Colin
will equalize them:
-3 + 5x = 3 - 3x
x=.75
So Colin plays (.75)CA, (.25) CB, and Rose wins on average ≤ ¾ .
Remember that Colin must do this randomly. He might achieve this strategy
any number of ways, but one way would be flipping two coins and playing B if he
gets two heads.
Now we can analyze this from Rose's point of view. We use the same
calculations, but with probability y:
ColinA: y(2) + (1-y)(0) = 2y
ColinB: y(-3) + (1-y) (3) = 3- 6y
2y= 3-6y
y= 3/8
So if Rose plays (3/8) RA, (5/8)RB, she is assured ≥ ¾ per game
This is similar to a saddle point. Rose has a mixed strategy, which ensures
she will have an expected payoff of ≥ ¾ . Colin has a mixed strategy which ensures
Rose's expected payoff ≤ ¾ . So ¾ is the expected value of the game, and ¾ CA, ¼ CB
is Colin's optimal strategy, while (3/8) RA, (5/8) RB is Rose's optimal strategy. This
is the solution of the game.
Application: Evolutionarily Stable Strategies
The basic ideas of game theory can also be applied to situations in which the
players are not making conscious choices or decisions; that is, game theory can
model different forms of behavior. The analysis of this can give insight into which
forms of behavior are “successful,” or have a tendency to persist, and which are not
and have a tendency to be driven out by others. This approach is very well suited to
evolutionary biology.
Evolutionary biology emphasizes the way that an organism’s genes are
responsible for its characteristics; an organism’s characteristics determine its
fitness and ability to produce offspring. An organism that is more fit in a given
environment is more likely to produce a larger amount of offspring, which increases
their representation in the population; these genes are carried on in the offspring, so
the organism’s offspring also has a greater rate of reproduction.
Evolutionary game theory comes into play when we understand that
behaviors of organisms involve interaction with other organisms, and an organism’s
success depends on this interaction. In order to analyze the fitness of an organism, it
must be analyzed in the context of its population. Thus, we can apply game theory in
this way: an organism’s characteristics and behaviors, determined by its genes, are
like its strategy in a game, and its fitness is its payoff; thus, the payoff depends on
which strategies (or characteristics) the organism uses (or possesses) in its
Jennifer Bachand
Math 482H
interactions with other organisms. Finding equilibrium is a helpful way to predict
the results of evolution on a population.
For example, we could consider a species of beetle. Suppose that the ability of
each beetle to find food and use this food effectively is determined ultimately by its
body size. Suppose further that a mutation is introduced: beetles with this mutation
develop a significantly larger body size. Our population then separates into two
separate kinds of beetles: small beetles and large beetles. Large beetles have higher
metabolic requirements, and thus require more food to maintain their energy; this
decreases their fitness.
We may think that because of this mutation, large beetles are likely to starve
and will therefore die out after several generations. However, we must consider also
the interaction between organisms: the beetles compete for food, and larger beetles
are more effective at claiming the largest shares of the food.
For simplicity, we will consider only the interactions between two beetles at
any point in time. However, note that the basic principles we will discuss can be
extended to simultaneous interactions among many beetles.
During a competition between two beetles for food, we have the following
outcomes:
• Beetles of the same size compete. Here, they get equal shares of the food.
• Beetles of different sizes compete. Here, the large beetle gets the majority of
the food.
This brings us back to the topic of evolutionarily stable strategies. We can
say that large beetles have less of a fitness benefit from a certain amount of food due
to their metabolism. We can model the beetles’ fitness from any amount of food as
payoffs in a two-player game between two beetles. Each beetle plays one of two
strategies, Small or Large, depending on its body size. Table 9 displays this payoff
table:
Beetle 2
Beetle 1
Small
Large
Small
Large
8,1
3,3
5,5
Table 9: The Body Size Game
1,8
These payoffs model the principles discussed: two small beetles will receive
the same fitness from the food source, large beetles receive more fitness at the
expense of the small beetles, and large beetles do not fully receive the fitness from
the food, since they have a higher metabolism and also must spend energy fighting
the other large beetle for an equal share.
This game differs from the ones previously discussed primarily because the
strategy played is not a choice. Each beetle, based on its genetics, is required to play
one strategy its entire life. Thus, the strategy changes only come from long-term
shifts in populations and the effects of evolutionary forces. This kind of game theory
is known as evolutionary game theory: players’ strategies are genetically
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Math 482H
determined and will persist once they are sufficiently introduced into the
population.
We suppose that each beetle will, throughout its lifetime, repeatedly compete
with other beetles for food. We also suppose that the population is big enough to
prevent the same two beetles from interacting solely with one another repeatedly
over their lifetimes. Carrying the game analogy further, we can say that a beetle’s
overall fitness is the average fitness it gains from each of its pairwise competitions
with others, and its overall fitness determines its reproductive ability and success.
This success is manifested by the amount of offspring carrying its genes into the
next generation.
A strategy is evolutionarily stable if its widespread use over the population
means that any small group of intruders adopting a different strategy will not be
successful, and will die off over several generations. These intruders could be
mutants born with new behaviors, or migrants who join the population.
Mathematically, we can say that whenever the entire population is playing strategy
S, a small group of intruders playing strategy T will have lower fitness than those
playing strategy S (or, strategy S dominates strategy T). Since fitness determines
reproductive success, it only makes sense that players using strategy T, as in a
subpopulation, will less successful reproductively and the amount of players of this
strategy will shrink over time and probably die out over multiple generations.
Thus, we have carried the game analogy this far:
• An organism in a population has fitness, which is the expected payoff it
receives from interactions with random members of the population.
• Strategy T invades strategy S at level x (x small and positive) if an x
percentage of the population plays T and (1-x) percentage plays S.
• Strategy S is evolutionarily stable if there exists a number y (y small,
positive) such that when any strategy T invades strategy S at a level x<y,
strategy S dominates strategy T; that is, an organism playing S has a strictly
greater fitness than an organism playing T.
We can apply these definitions to our beetle example. We will analyze whether
the Small strategy is evolutionarily stable, then we will analyze the Large strategy.
Suppose that for some small positive number x, a (1-x) fraction of the
population plays Small, and an x fraction plays Large; this models the situation
where a minor intruding population of large beetles has just arrived into a
population of small beetles.
The expected payoff to a small beetle in a random interaction, or game, in this
population, is determined as usual. Its payoff is 5 with the probability (1-x) and 1
with a probability x (from meeting a small beetle and a large beetle, respectively).
Thus,
E(x) = 5(1−x)+1·x = 5−4x.
The expected payoff to a large beetle in a random interaction in this population
is as follows: the large beetle receives a payoff of 8 with a probability of (1-x) and a
payoff of 3 with probability x (from meeting a small beetle and a large beetle,
respectively).
Thus,
Jennifer Bachand
Math 482H
E(x)= 8(1−x)+3·x = 8−5x.
For small values of x, or even relatively larger ones, E(x) for the fitness of the
large beetles is larger than the E(x) for the fitness of small beetles. This tells us the
the Small strategy is not evolutionarily stable.
What about the Large strategy? We can now suppose that for some small,
positive number x, a (1-x) fraction of the population plays the Large strategy and an
x fraction of the population plays Small. This models the situation where a minor
intruding population of small beetles has just arrived into a population of large
beetles.
The expected payoff to a large beetle in a random interaction in this population
is as follows: the large beetle receives a payoff of 3 with probability (1-x) and a
payoff of 8 with probability x (from meeting a large beetle and a small beetle,
respectively). So
E(x)= 3(1−x)+8·x = 3+5x.
The expected payoff to a small beetle in a random interaction in this
population is as follows: the small beetle receives a payoff of 1 with probability (1-x)
and a payoff of 5 with probability x (from meeting a large beetle and a small beetle,
respectively). So
E(x) = (1−x)+5·x = 1+4x.
We can see that E(x) for large beetles is greater than E(x) for small beetles, so
the Large strategy is evolutionarily stable.
These results tell us that if a small population of large beetles is introduced
into a population of small beetles, the large beetles are very successful; since they
mostly meet small beetles, the get most of the food in almost every interaction they
have. Since the small beetles cannot compete with the large beetles, Small is not an
evolutionarily stable strategy.
Conversely, if a small population of small beetles is introduced into a
population of large beetles, the small beetles will still lose in almost every
competition for food. The population of large beetles is able to resist the intrusion of
small beetles; Large is an evolutionarily stable strategy.
Therefore, if a mutation to create larger body sizes is possible, we should
expect that only populations of large beetles will exist in nature, as they will be
evolutionarily stable; large beetles benefit at the expense of small ones.
There is another way to interpret our results. Notice that small beetles receive
more fitness from the food that they ingest than the large beetles. We can say that if
we started from a population of small beetles, natural selection caused these
beetles’ fitness to decrease over time as larger beetles were able to do better in
interactions with them (a payoff of 5 turned into a payoff of 1, decreasing the small
beetles’ fitness). This fits with the main principle of natural selection: natural
selection increases the fitness if individual organisms in a fixed environment, but if
the environment becomes more hostile to the organisms, as in the case where a
population of large beetles invades, the organism’s fitness will intuitively go down.
This is the case with the beetles; the increasing amount of large beetles can be
seen as a shift to an environment which is more hostile for everyone. Only the fittest
will survive.
Jennifer Bachand
Math 482H
References:
1. Straffin, P. D. Game Theory and Strategy. The Mathematical Association of
America, 1993. Print.
2. Ventzel, Elena S. Lectures on Game Theory: International Monographs on Advanced
Mathematics and Physics. Delhi: Hindustan Publishing Corporation, 1961. Print.
3. Easley, David, and Jon Kleinberg. Networks, Crowds, And Markets, Reasoning About
A Highly Connected World. Cambridge University Press, 2011. Print.
4. Vajda, Steven. Mathematical Games And How To Play Them. Chichester: Ellis
Horwood Limited, 1992. Print.