Teacher professional development and classroom resources across the curriculum
Teacher professional development and classroom resources across the curriculum
In the Prisoner's Dilemma, the players have a choice of whether or not to cooperate with one another or to defect. We saw that what the players should do—that is, their best strategies-depend on whether they will be playing just once or many times. If they are to play only once, they should both defect, even though it would be better overall for them to cooperate. If they are to play many times, however, it behooves them to try different mixes of cooperation and defection. Axelrod's tournament study demonstrated that, over the course of numerous games, players who play pure strategies can be beaten by players who choose mixed strategies.
We can extend this thinking to the natural world if we imagine the competition for survival to be a tournament. The many rounds of this tournament correspond to the daily struggles that certain species face for survival. In the natural world, all species compete for resources using wildly varying strategies. The strategies that are most successful in the long run are the ones that survive to be passed along to offspring.
Strategies for survival in the natural world do vary, but broad trends are discernible. For our purposes, we will simplify things greatly by limiting the options to only two types of behavior, aggressive and passive, and we'll call the actors of these behaviors "Hawks" and "Doves" respectively. Aggressive animals will always fight over resources, whereas passive animals will not. This is the basis for a famous game often known as "Hawks and Doves," which was first proposed by John Maynard Smith and George Price in a 1973 paper.
The assumptions behind the game are pretty straightforward. Imagine a field strewn with piles of food. This field is populated with animals that can behave either passively or aggressively toward one another. The animals, or players, compete with one another for the resource piles. For simplicity's sake, we determine that all competitions occur between individuals, i.e., one-on-one. The standard scenario is that one animal approaches a pile of food, and then another animal presents a challenge for it. Furthermore, neither animal knows the other's behavioral identity until the challenge has begun. The possible interactions are then Hawk-Hawk, Hawk-Dove, Dove-Hawk, and Dove-Dove.
Whenever a Hawk fights another Hawk, one of them wins the entire food pile, thus getting a benefit, B. The loser gets injured, incurring a cost, C. Both B and C can be thought of as food calories. The resource calories gained by the winner are counted as positive, but the calories that the loser must devote to healing are counted as negative. For the purposes of this game, we assume that all Hawks are equal and win half of all their battles with other Hawks. Also, as with the Prisoner's Dilemma game, we are concerned only with the tendencies established through iterative scenarios. Consequently, on average, a Hawk will gain calories and lose calories in a Hawk-Hawk interaction. This average energy accounting can be simplified to .
When a Hawk challenges a Dove, the Dove does not fight but simply walks away. This means that the Hawk gets the entire benefit, with no cost of fighting. The Dove gets nothing, but also loses nothing. In Hawk-Dove and Dove-Hawk interactions, the Hawk always gets the entire benefit, B, and the Dove always gets nothing and loses nothing. Therefore, the Hawk's average payoff is B, and the Dove's is zero.
Finally, when a Dove challenges a Dove, they do not fight but, rather, split the resource evenly. Each player gets without any cost to anyone.
Notice that this is not a zero-sum situation, because not all of the cells add up to the same value; the Hawk-Hawk interaction yields less in total benefit than the other three scenarios.
Now that we have a grasp of the basic circumstances of the game, let's think about whether it's better to be a Hawk or a Dove. It might seem, at first glance, that being a Hawk is always the best idea. If we imagine that the population of the field is nearly 100% Hawk, it's hard to see how a Dove could ever survive for very long, as it would get to eat only upon encountering another Dove. On the other hand, if the field is nearly 100% Dove, then a single Hawk is going to have it incredibly easy. This would lead us to think, if we had to choose between playing Hawk and playing Dove, that we should always choose Hawk. After all, a Hawk in an all-Dove world is going to do well, whereas a Dove in an all-Hawk world is going to starve.
That's the standard intuition, but let's consider the situation of the lone Dove a little more carefully. He never loses calories, and while the Hawks are gaining calories, they are also losing them in their fights. If these costs end up being more than the resource benefits, then each Hawk will experience an overall calorie loss as time goes on, while the Dove holds steady (this assumes, of course, that there is no cost for simply waiting around while everybody else fights amongst themselves). After a while, the lone Dove will be doing much better than the always-fighting Hawks. This suggests that if costs are more than benefits, one might do well to be a Dove in an all-Hawk world.
If Doves do better in an all-Hawk environment when costs outweigh benefits, then over time the population should shift toward all Doves. This is based on the assumption that the most fit, the ones with the highest net calories, survive to reproduce more often than the less fit.
One might then think that whenever costs outweigh benefits, the population will tend to evolve into all Doves. However, a Hawk in an all-Dove environment will do extremely well relative to the Doves, even if costs outweigh benefits. This is because the cost becomes irrelevant if there are no other Hawks around to inflict injuries. We would then be led to believe that the all-Dove scenario is not stable, even when costs outweigh benefits.
The idea of a pure strategy's stability is an important one. In our analysis, we saw that neither the all-Hawk nor the all-Dove strategy is stable when the costs outweigh the benefits. This means that either situation can be infiltrated by the opposing strategy. Note that this is not true when the benefits outweigh the costs. Such a world would be driven towards the all-Hawk state, as a lone Dove would gain nothing while the Hawks gained something from each fight. This suggests to us that the relationship between costs and benefits has something to do with which state will be stable. Furthermore, we can conclude that because neither the all-Hawk nor the all-Dove state is stable, if there is to be a stable state, it must lie somewhere between the pure states. This means that if one has a choice as to whether to be a Hawk or a Dove, it would be best to adopt a mixture of the strategies-but what mixture? Remember that on the level of each individual confrontation, you have to choose your identity, whether to be a Hawk or a Dove, before you know the identity of your opponent. What percentage of the time should you be a Hawk and what percentage of the time should you be a Dove? With just a little algebra, we can find these percentages:
To start, let's represent the pure Hawk strategy as H, the pure dove strategy as D, and the Mixed strategy as S. The payoffs for these would be as follows:
E(H,S) = payoff of pure Hawk versus the Mixed strategy
E(D,S) = payoff of pure Dove versus the Mixed strategy
Let's define p as the probability that the Mixed player plays Hawk in a given interaction; then the expression 1-p represents the probability that the Mixed player will play Dove. The expected average payoff of H vs. S, E(H,S), will be composed of part of the Hawk-Hawk and part of the Hawk-Dove payoffs.
E(H,S) = (probability that S plays Hawk) x (payoff of Hawk-Hawk) + (probability that S plays Dove) x (payoff of Hawk-Dove)
The expected average payoff for D vs. S, E(D,S), can be found in a similar manner:
E(D,S) = (probability that S plays Hawk) x (payoff of Dove-Hawk) + (probability that S plays Dove) × (payoff of Dove-Dove)
S's optimum mix will be when both H and D do equally well against it. This means that S has nothing to gain by skewing the mix towards more Hawk or more Dove than prescribed by p and (1-p) respectively. In other words, the optimal mix will be the value of p when E(H,S) = E(D,S).
Solving this for p yields the percentage of time that S should play Hawk, which turns out to be . Note that this percentage is entirely dependent on the benefit-to-cost ratio.
All of this means that were we to study the population of our field for a long time, we would find that the ratio of benefits given by food piles to costs incurred by fighting would determine the percentage of time that a Mixed animal should play Hawk or Dove. If, for some reason, the system falls out of balance, as when a group of players decides to play Hawk more often than they should, then there will be a clear advantage for others to play Dove more than they should. These counteracting forces would then drive the system back to the appropriate average ratio of Hawks to Doves.
The evolutionary progress of our field, like the process of Axelrod's tournament, shows that pure strategies are neither always stable, nor always optimal. The most successful strategies are usually mixed strategies. In terms of human behavior, this suggests that to be successful, we should not be too quarrelsome, nor should we be pushovers. Additionally, we should be forgiving at times, and at other times we should not hesitate to retaliate against wrongdoers. These conclusions are all well and good in theory, but how do they play out in real life with actual human beings? In our next section, we will examine what happens when game theory's predictions are put to the test in different human cultures.