Matrix games

We can identify pure strategy ESSs directly from the payoff matrix.

The numbers on the diagonal of the matrix are the payoffs for each strategy against itself.

Since a pure strategy ESS is the best response to itself, the number on the diagonal for that strategy will be larger than any of the other values in that column (since these describe how other strategies do against that strategy).

In the second example above, S₁ is a pure strategy ESS since 1 > 0. In the third case, S₂ is a pure strategy ESS since 2 > 1. In the fourth case shown, both strategies are pure ESSs, since both diagonal elements are greater than the other elements in their respective columns.

Note that in the first example shown, there is no pure strategy ESS, since the diagonal elements are each the smallest element in their respective columns.
For a 2 strategy game, this means that there is a mixed strategy ESS at which both S₁ and S₂ persist.

For the case of 3 or more strategies, there may be no ESS at all, so we can not infer the existence of a mixed strategy ESS from the lack of pure ESSs.
An example of such a game with no ESSs at all is "Rock-paper-scissors"

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Homework problem

In the Hawk-Mouse game from the previous lecture, consider these two cases:
Case 1: B = 1.4, C = 1.2
Case 2: B = 1.4, C = 1.6

For each of these cases:
1) Write the equations for W_H and W_M for these values of B and C.
(See Equations 1 and 2 from the previous lecture.)

2) Plot W_H and W_M against the frequency of Hawk (p).

3) Write the payoff matrices for both cases.

4) Identify any ESSs.

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Evolution of cooperation

How can organisms evolve to cooperate with one another even when it seems that they would do better by not cooperating?

Such cooperation is often seen even between individuals that are not related to one another.

The problem is that, in a single encounter, cheating ("defection") tends to do better than cooperation.

Simple case: the Prisoner's Dilemma
Consider two prisoners being interrogated separately

Both remain silent (cooperate with one another) - both get moderate sentence (1 year)
You squeal (defect), the other doesn't - you go free
Other squeals, you don't - you get harsh sentence (10 years)
Both squeal - both get 5 years

We can write a matrix showing your "payoff" as a function of what you do and what your partner does:

Note that, whatever your partner does, you always do better by defecting than by cooperating.

Since your partner has exactly the same options and payoffs, they will also conclude that their best strategy is to defect.

Thus, in a single game, if both players use their optimal strategy, they end up both getting 5 years (instead of only 1 year, which would have been the result if they cooperated).

Example: Bacteriophage.

Viruses carry genes that produce proteins that allow them to usurp their host's DNA replication machinery in order to replicate themselves.

When more than one virus infects a particular host cell, the viral proteins produced by each are available to all, since they are free in the cytoplasm.

As a result, an individual virus particle can "cheat" by not producing its own viral proteins (or producing less) and using those produced by other viruses that have infected the same cell.

An example of this has been observed with the bacteriophage 6, which infects Pseudomonas bacteria. A variant of 6, called H2, produces fewer viral proteins than does 6.

The payoff matrix when these two phage strains interact looks like this:

6 H2

6 1 0.65

H2 1.99 0.83

In laboratory studies in which bacterial cells are each infected with many phage, H2 sweeps through the population (as expected from this payoff matrix), leading to reduced .

This does not occur when each bacterial cell is infected with only a single virus particle.

Repeated prisoner's dilemma game.
Here, the two players meet more than once, with the potential for each to remember past encounters.

in a "tournament" that played many different strategies against one another, the winner was the simplest strategy proposed:
-----"Tit-for-tat" strategy = Cooperate the first time, then do whatever your opponent did last time you met.

Tit-for-tat (or TFT) is an ESS against "Always defect" (AllD).
However, AllD is also an ESS against TFT.

This is because each strategy does poorly when it is rare, but well when it is common.

The reason that Tit-For-Tat (TFT) is an ESS against Always Defect (AllD) is that TFT does particularly well playing against itself, since both players repeatedly cooperate. It thus changes one element in the payoff matrix relative to the single Prisoner's Dilemma game.

Note that AllD is still a pure strategy ESS as well, since it does better against itself than TFT does against it.

Reciprocal altruism
An individual may help another, at a short term cost to itself, if there is sufficient chance that the other individual will later repay the favor.

Requires that individuals interact over a long time, recognize one another, and can remember past interactions.

Example 1: Vampire Bats
These roost together in social groups, but forage alone.
If a bat fails to feed, its roost mates that were successful will regurgitate some blood for it.

This is common only among bats that normally roost together and thus recognize one another.

Example 3: Pied Flycatchers
When these birds are threatened by a predator such as an owl or hawk, they "mob" it. This means making a distinctive call to attract other flycatchers who then attack the predator as a group.
This is altruistic behavior, since mobbing birds are sometimes injured or killed by the predator.

In an experiment (using a fake owl as a predator), when one flycatcher is prevented from answering a mobbing call from another (because researchers caught and caged it), the other will not come to help it if it is subsequently menaced by predator. Jul 8, 2021