Conflict in Loons, Continued
Overview
This tutorial shows how to build an asymmetric game. It assumes that you have already completed the Conflict I, Probability in Payoffs, and Stable Strategies tutorials and have a good understanding of payoffs and symmetric games. If you have not yet done those tutorials, you should go through Conflict I before continuing here. If you work through all examples in detail, this tutorial should take about 45 minutes. This tutorial also refers to the tutorial on Nash Equilibria (~30 minutes).
Introduction
In the Conflict I tutorial, you built a model of conflict over territories in loons. At the end of that tutorial, you learned that the model leaves out an important point — that a territory is more valuable to a resident male than it is to an intruding male.
The resident male has usually held his territory for several years and has found the best nesting site on it. When a new male takes over, he goes through that process again himself. This could take several seasons of trial-and-error, during which breeding may be unsuccessful.
How can we build a model that takes into account the difference in territory value to residents and intruders? There are different payoffs for residents and intruders, so we need a separate payoff matrix for each role, resident and intruder.
Symmetric games are those in which both players have the same strategies and get the same payoffs for them.
Asymmetric games are ones in which players have different strategies or get different payoffs for their strategies.
Asymmetric Hawk-Dove
Using just the Hawk and Dove strategies for now, this is what the matrices for loon conflict look like. As usual, each cell of the payoff matrix contains the payoff to the strategy on the left when played against the strategy above. In the matrix for residents, all payoffs are against intruders, while in the matrix for intruders, all payoffs are against residents.
| Hawk (Intruder) | Dove (Intruder) | |
|---|---|---|
| Hawk (Resident) | payoff to resident hawk vs. intruding hawk | payoff to resident hawk vs. intruding dove |
| Dove (Resident) | payoff to resident dove vs. intruding hawk | payoff to resident dove vs. intruding dove |
| Hawk (Resident) | Dove (Resident) | |
|---|---|---|
| Hawk (Intruder) | payoff to intruding hawk vs. resident hawk | payoff to intruding hawk vs. resident dove |
| Dove (Intruder) | payoff to intruding dove vs. resident hawk | payoff to intruding dove vs. resident dove |
To show the logic of an asymmetric game, we gave each role its own payoff matrix above. In most textbooks, the two sets of payoffs are combined into one matrix and separated by commas as follows, and we will generally follow this convention from now on:
| Intruder | |||
|---|---|---|---|
| Hawk | Dove | ||
 |
Hawk | resident hawk vs. intruding hawk, intruding hawk vs. resident hawk |
resident hawk vs. intruding dove, intruding dove vs. resident hawk |
| Dove | resident dove vs. intruding hawk, intruding hawk vs. resident dove |
resident dove vs. intruding dove, intruding dove vs. resident dove |
|
The payoff on the left side of the comma is for the strategy and role to the left against the strategy and role above. The payoff on the right side of the comma is for the strategy and role above against the strategy and role to the left.
What is the simplest way to modify the old Hawk-Dove game to take into account the extra value that a territory has to its resident? Recall that the Hawk-Dove game has two variables, v (the value of the resource) and c (the cost of losing a fight), and the following payoffs.
| Hawk | Dove | |
|---|---|---|
| Hawk | v/2−c/2 | v |
| Dove | 0 | v/2 |
This symmetric game can be the basis for a new asymmetric game. Before filling in the payoff matrices, we need a new variable for the extra value that the territory has to its resident male. Let’s call it k, for the local knowledge held by the resident. Now the value of a territory for its resident is v + k, while the value of a territory to an intruder is v. (We could instead have chosen to have two different value variables, one for the the resident and one for the intruder. It makes no difference.)
First, let’s fill in the payoffs for the intruder:
| Intruder | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Hawk | Dove | ||||||||
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Hawk |
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| Dove |
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That was easy — it’s exactly the same as the standard symmetric Hawk-Dove game! It makes no difference that the resident values the territory more than the intruder, because we were only concerned with the intruder’s payoffs. Now let’s fill in the payoffs for the resident:
| Intruder | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Hawk | Dove | ||||||||
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Hawk |
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| Dove |
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The resident’s payoffs are very similar to those in the symmetric Hawk-Dove game; the only difference is that the resource value is (v+k) instead of v. Here are the resulting payoff matrices for this asymmetric game, separated by role for clarity:
| Hawk | Dove | |
|---|---|---|
| Hawk | (v+k)/2−c/2 | v+k |
| Dove | 0 | (v+k)/2 |
| Hawk | Dove | |
|---|---|---|
| Hawk | v/2−c/2 | v |
| Dove | 0 | v/2 |
Experiment with this model in the game simulator below. Try different values of v, k, and c. Try starting with a population in which all the residents and intruders are doves, then introduce the Hawk strategy to one role or the other by changing its initial proportion from 0 to 1. With what relative values of v, k, and c does Hawk become the dominant resident strategy? How does increasing c affect residents and intruders differently? Next, try changing the initial proportions of Hawk and Dove independently for each role (Resident and Intruder); how does this affect the outcome?
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It’s not a complete explanation, but this game provides some insight. In the symmetric game, Hawk was dominant when v > c, and that is the case for both the Intruder and Resident roles here as well. However, when v < c (which gave a mixed ESS in the symmetric game), Hawk can dominate in either the Intruder or the Resident population, with the other role playing Dove. The outcome is determined by the initial proportions of Hawk and Dove in each role. However, when v < c and k > 0, there’s a complex interaction between the values of v, c, and k and the initial proportions of Hawk and Dove in each role. It’s only when v + k > c and c > v that we see a clear result — residents play Hawk and intruders play Dove regardless of the initial proportions. (Try it; set v = 9, c = 12, k = 3.5, then set the proportions of resident Hawk = 1, resident Dove = 100, intruder Hawk = 100, and intruder Dove = 1. You may need to increase the number of generations to see the results.) Given the high cost of losing and the high added value of a territory for its owner, it appears that a resident male is more motivated than an intruder to fight, exposing himself to the risk of injury.
There are two problems with that interpretation. First, a resident male playing Hawk risks injury only if the intruder also plays Hawk, and the model shows that intruders should instead play Dove when v + k > c and v < c. Second, we’ve already established that intruders regularly fly over territories, presumably to assess resident males. It is more likely that they play Assessor than Dove. In fact, as you saw in the Conflict I tutorial, in symmetric games, Assessor is the only evolutionarily stable strategy, always beating Hawk and Dove.
You may have noticed that this simple-looking game has rather complex behavior. Which strategy becomes dominant depends on the initial proportion of each strategy in each role. It is not as easy to empirically find evolutionary stability as it was in the symmetric Hawk-Dove game. Go back to the simulation and try different initial proportions of Hawk and Dove for resident and intruder males. Can you make any generalizations about what initial conditions lead to what outcomes?
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Stable States
In a symmetric game, you can calculate the evolutionarily stable strategy (ESS). In an asymmetric game, there are two roles with different strategy sets, so stability consists of a pair of strategies, one for each role. The stable state in an asymmetric game is called a strict Nash equilibrium. After completing the Nash Equilibrium tutorial, continue here by finding strict Nash equilibria in the asymmetric Hawk-Dove game (the link opens in a new window so you won’t lose your place in this tutorial).
| Intruder | |||
|---|---|---|---|
| Hawk | Dove | ||
|
Hawk | (v+k)/2−c/2 , v/2−c/2 | v+k , 0 |
| Dove | 0 , v | (v+k)/2 , v/2 | |
Remember that v is always > 0, k is always ≥ 0, and c is always ≥ 0. You’ll need paper and pencil to work these out, but you can check your answer with the ✓ button after each one.
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If those calculations didn’t seem straightforward, it may be because working with inequalities is unfamiliar. That will improve with practice, so the main thing is to be sure that you understood how to check each cell of the matrix to determine whether it was a Nash equilibrium. Now that you’ve seen the answers, you may want to go back to the Asymmetric Hawk-Dove simulation to see if they make sense empirically.
You may have noticed that the Hawk-Dove and Dove-Hawk equilibria overlap. Hawk-Dove is an equilibrium when c > v, while Dove-Hawk is an equilibrium when c > v+k. But when c > v+k, c is also greater than v, so both conditions are met (e.g. c = 10, v = 8, and k = 1). In that case, both Hawk-Dove and Dove-Hawk are equilibria, but which one the game stabilizes at depends on the initial proportions of Hawk and Dove in each role. So Dove-Hawk is never an equilibrium on its own. On the other hand, Hawk-Dove is the only equilibrium when c > v and c < v+k (e.g. c = 10, v = 8, and k = 4).
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Asymmetric Assessor
Of course, even knowing the Nash equilibria, the asymmetric Hawk-Dove game can’t fully explain the loon-fighting data. We need to add the Assessor strategy, which judges its opponent. If its opponent is larger, an Assessor acts as a Dove. If its opponent is smaller, an Assessor acts as a Hawk. This is what the symmetric Hawk-Dove-Assessor game looks like (go back to the Conflict I tutorial if you’ve forgotten where these payoffs come from):
| Hawk | Dove | Assessor | |
|---|---|---|---|
| Hawk | v/2−c/2 | v | v/2−c/2 |
| Dove | 0 | v/2 | v/4 |
| Assessor | v/2 | 3v/4 | v/2 |
Let’s turn this into an asymmetric game with our new variable, k, for the added value that a territory has for its resident male. Again, we need two sets of payoffs, one for Resident and one for Intruder. As in the asymmetric Hawk-Dove game, payoffs for Intruder are the same as in the symmetric game, so you can go straight to the Resident payoffs. Payoffs between Hawk and Dove are as before; we need new ones only when Assessor is involved.
| Intruder | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Hawk | Dove | Assessor | |||||||||||
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Hawk |
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| Dove |
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| Assessor |
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Explore this game in the simulator. Can you find any Nash equilibria by trying different values for v, k, and c? How do the initial proportions of each strategy affect the outcome over time?
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Stable States
What are the strict Nash equilibria in this game? There are 9 strategy pairs to consider, but you can use a shortcut to make it easier. Remember that for a strategy to be stable, its payoff must be greater than for any other strategy, and this must be true for both roles if the pair is stable. So if you can find even one equal or better payoff in the column of payoffs to the left of the comma, you needn’t consider any further — this pair is not stable. Similarly, if you find even one equal or better payoff in the row of payoffs to the right of the comma, the pair is not stable. So the quickest thing to do when considering a cell in the payoff matrix is to look for an obviously equal or better payoff in either the column or row. Only if you don’t find one do you need to consider the actual inequalities.
For example, Hawk-Hawk was a strict Nash equilibrium when v > c in the asymmetric Hawk-Dove game. Is it stable here? Look at the resident’s payoffs in the Hawk column. Is (v+k)/2−c/2 the best payoff? No, the Assessor payoff of (v+k)/2 is better, so Hawk-Hawk is not stable and there is no need to look at it further. (You could instead look at the intruder’s payoffs in the Hawk row and ask if v/2−c/2 is best. It is not, because the Assessor payoff of v/2 is better.)
Now try it yourself. It should go fairly quickly using the shortcut. Remember that v is always > 0, k is always ≥ 0, and c is always ≥ 0.
| Intruder | ||||
|---|---|---|---|---|
| Hawk | Dove | Assessor | ||
|
Hawk | (v+k)/2−c/2 , v/2−c/2 | v+k , 0 | (v+k)/2−c/2 , v/2 |
| Dove | 0 , v | (v+k)/2 , v/2 | (v+k)/4 , 3v/4 | |
| Assessor | (v+k)/2 , v/2−c/2 | 3(v+k)/4 , v/4 | (v+k)/2 , v/2 | |
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So Assessor-Assessor is the only strict Nash equilibrium in this game. That’s not too surprising, given that Assessor was also the ESS in the symmetric conflict game. Now that you know this, does it help explain the loon-fight data?
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What’s Missing?
A full explanation for loon fights remains elusive. If both the resident and the intruder play Assessor, there should never be any injuries, since the smaller male should always concede without a fight. Of course, on those rare occasions when the resident and intruder are evenly matched, two Assessors would fight, and the fights could be long and intense. This matches the data. The difficulty lies in explaining why only resident males are injured enough to die in a fight. The resident seems to play Hawk (risking injury) while the intruder plays Assessor (fighting only when he is likely to win). Can we explain this?
An Assessor judges its opponent and acts like a Dove when the opponent is stronger but like a Hawk when the opponent is weaker. What if residents, but not intruders, tend to overestimate their own strength? If so, residents would sometimes engage in fights that they cannot win. Why might a resident overestimate his own strength? As pointed out in the Conflict I videos, a resident male is constantly subject to harassment by intruding males. He is so busy watching for and warding off intruders that he is unable to feed himself and loses weight over the course of the breeding season. Thus his strength declines over time, but his self-assessment could be out-of-date.
Another possibility is that an intruder has better opportunities for assessment than does a resident. Our description of the Assessor strategy was vague about the nature of assessment. It could be visual, with intruders and residents judging each other’s relative sizes. It could be auditory, based on the pitch and loudness of calls made by a resident to deter an intruder who flies overhead (Charlie Walcott has evidence for this). It could include small skirmishes, tests of fighting ability that may not escalate into full fights. It might take place only on initial contact, or it might continue during a fight. Assessment is likely to involve all of these interactions. Perhaps an intruding male, by flying over breeding territories and observing residents, has a better opportunity to assess a resident than a resident has to assess an intruder. Indeed, a resident may have to make his evaluation on the spot when challenged, while the intruder evaluates the resident over time. It seems likely that an intruder has better information about the resident’s condition than the resident has about the intruder’s. This may lead to errors that cause a resident to fight when he shouldn’t.
Finally, what about female loons, who fight but do not get seriously injured? It seems likely that they play a symmetric game with both playing Assessor. Resident and intruding females probably value the territory equally, given that the male is responsible for finding a nest site. That is itself mysterious. Why does a resident female not use her local knowledge when a new male takes over? Furthermore, if her reproductive success is tied to the resident male’s knowledge of the best nesting site, why does she not assist him against an intruding male, who is less valuable as a mate because he lacks that knowledge?
Keep in mind that these models only describe one aspect of loon behavior. They are also involved in other games (e.g. parental care games with their partners), and the outcomes of all these games interact to determine each loon’s fitness. It would be a mistake to expect one model to explain all aspects of any animal’s behavior. Our goal should be to explain one aspect as thoroughly and simply as possible and then use the model to generate hypotheses to be tested with more data.
Further Exploration
Here are some additional questions to consider about asymmetric conflict:
- What predictions does the asymmetric Assessor game make? How might they be tested in the field?
- In human conflict, what are some different ways in which asymmetry might arise?
- How would you change the asymmetric Hawk-Dove game to give the intruder a greater probability of winning a fight?
This concludes the tutorial on asymmetric conflict. You may now try the practice questions for asymmetric games (not yet available).
For Cornell BioNB 2210 students: download the Game Theory Quiz here (the same quiz is posted at the end of Conflict I, Conflict II, and Cooperation; we urge you to complete all three tutorials before answering the quiz).