One day you are walking along the street, and someone in front of you drops a $50 bill. Even though no one around is really paying attention, most of us will pick it up and give it back to the person. Why is that?
Maybe it has something to do with morals in this case — we feel that it is good to return the money to the owner. But we can dig a little deeper and ask why we have any norms around ownership at all? If I ever drop something on the street like a phone or wallet, I’m not really concerned that someone will pick it up and take it before me, even if the person is clearly physically stronger and would have a decent chance in a fight over the item.
And even if this wasn’t enough to convince you that something more than our good-natured morality is going own, the fact that ownership is respected in baboon troops when meat is involved should tell us that there is more of a story here.
We have an account from Maynard Smith and Parker: by coordinating using some kind of asymmetry, agents can more efficiently divide up resources.
Consider, for example, a simple instance of the Hawk-Dove game given in the following table:
You can also think of this as the game chicken, where two people drive towards each other. The aggressive ones — hawks in the above table — never swerve. The passive ones — doves — always swerve. And then you can have people who are sometimes agressive and sometimes passive, choosing whether to swerve or not at chance. These people play what we call a mixed strategy.
If you are one of the players, you prefer for to drive straight and for the other person to swerve, thus getting all the glory, to the other three possibilities. This is reflected in the payoff matrix as a value of 4. The next best is for both of you to swerve, since you don’t die and the other person doesn’t get any glory either. This is reflected in the value of 2. The next best is for you to swerve when the other person goes straight since at least you are alive. This is reflected in the value of 1. The worst case is where both of go straight, and die in the crash.
Now take a second to think about what you would do if you had to commit to your action before you say what the other person would do (and they would have to do the same).
In this situation the best you can do is to play a mixed strategy. The exact probabilities of swerving versus driving straight depend on the specific payoffs (in this case the optimal probability is 1/2).
We can also think of this not as an individual player playing a mixed strategy, but as the distribuition of a population. For example, instead of a person swerving with probability 1/2 and going straight with probability 1/2, we can thinking of a population wherein half of the indviduals are hawks (always going straight) and half are doves (always swerving). This population is stable, in the sense that no individual has an incentive to switch their strategy.
Now let us return to ownsership priority. We can view this as an instance of the general phenomena that we can do better in the Hawk-Dove game (and other games) by conditioning our strategy on an assymtery. For example, in the chicken version of the game, suppose we had a convention that the shorter person would drive straight and the taller person would swerve (and assue we could always tell who was taller). Then this would allow us to always anticoordinate our strategies so that one of us would swerve and the other would drive straight.
So, if we are in a context in which ownsership is salient, such as the case with the $50 bill, then we can see the ownership priority convention as a way in which we exploit assymetries to (anti)coordinate our moves in the game. Instead of both fighting over the $50 and risking physical damage, the person just hands over the bill and we both go on our merry way.
Now, however, Eshel raises a question: why the ownership convention over any other? Indeed, why not the “paradoxical” strategy — the one in which the owner defers to the other?
***The original paper can be found here.***
This is an instance of the more general questions Eshel has: “Concerning conflicts within natural populations, a preliminary question to be asked is, therefore: What is the set of relevant asymmetries, expected to be observed by individuals in a given population?” (p. 3)
The idea is this: given a particular set of asymmetries that are considered relevant by the individuals in the population we can figure out how we expect the population to behave using the tools given to us by Maynard Smith and Parker. However, we also need to settle the issue of which asymmetries are considered relevant in the first place. We can’t use the same technique to solve this problem, so it seems that we are a little stuck. Indeed, Eshel’s main claim is that
none of the behavioral rules, observed in such populations, can be explained on the pure basis of population game theory. This is so because virtually almost any set of asymmetries, once established as relevant in a large enough majority of the population, can lead to an evolutionary stable situation”
Eshel, p. 4
Thus, if we do want to explain why we end up with something like ownership priority as opposed to something else, we have to appeal to non game-theoretic reasons. I think that this is a neat little observations about the limits of a particular set of tools.
The Crow's Nest 
Is this not simply a limitation of classic game theory compared to the more dynamic evolutionary models? Perhaps the player response or strategy set should be reconsidered.
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I’m not exactly sure what you mean by “this”, but I agree that expanding the strategy set can help — and I think that is exactly what we are doing when we add conditional strategies to the game. However we are also adding the asymmetries themselves, so we give the game a little more structure as well. I don’t think this is unique to dynamic evolutionary models though, you can have those in classical game theory as well. But in general you are right that when we encounter a limitation of our tools there are many ways we can try to expand our toolkit.
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