# The name of the game (Hawk-Dove variant?)

I stumbled upon the following symmetric two-person game. We have two objects $X,Y$ with positive value $x$ and $y$, and two persons that have to pick independently form each other simultaneous one of the objects. If a person is the only one that picks $X$, then she receives $x$ as payoff. If two persons pick $X$, then everybody gets $x/2$ as payoff. Some goes for $Y$.

This gives as payoff matrix $$\begin{pmatrix}x/2,x/2 && y,x \\ x,y && y/2,y/2\end{pmatrix}.$$

Does this kind of game have a certain name? It looks similar to Hawk-Dove or chicken, but it is different. It seems to be a very natural instance of a game (also its generalization for more players), so I wonder if this is known as a classic example in game theory.

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A relevant (inexact) analogue might be the battle of the sexes game, with one player's strategies switched. – Kevin Costello Nov 22 '12 at 19:31

This game describes a conflicting situation and it is clearly an anti-coordination game. The (pure) Nash equilibrium payoffs are $(x,y)$ and $(y,x)$. We can safely say that this is an example of Hawk-Dove game. However, note that not every Hawk-Dove game must be of that type. That is, it is not necessary to put a restriction like $x/2$, it could be something else as long as it is strictly less than $x$. (to assure $(x,y)$ is an equilibrium outcome)

P.S. This game is definitely not prisoner's dilemma. First of all, there is no dominant strategy in this game.

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I think this is a special case of the prisoners dilemma.

However, not in general, only if x>0 and y<0 (or vice versa).

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No, it is not. In the prisoner's dilemma you get the least payoff, when different strategies are used. In the game I ask for, the minimum payoff is generated when the two players pick both the smaller-valued object. – A.Schulz Nov 16 '12 at 12:50
I would say it differs from the prisoner's dilemma the following way: The best outcome is a Nash equilibrium. The "dilemma" in prisoner's dilemma is that this is not the case. "Choosing the same strategy" and "choosing the opposite strategy" isn't really viable qualifiers. Imagine if they sit opposite eachother and decide whether they want the one on their left or right. Choosing different strategies in that instance is the same as both choosing $X$ in your formulation of the game. – Arthur Nov 16 '12 at 12:58
I disagree again. Renaming the strategy for one player corresponds to a column or row exchange in the payoff matrix. If you swap the columns in my payoff matrix than the form is also not prisoner's dilemma. See en.wikipedia.org/wiki/Prisoner%27s_dilemma for a definition of pd. – A.Schulz Nov 16 '12 at 15:48
Edited the post to be more accurate, note that it is only a partial answer. – Dennis Jaheruddin Nov 16 '12 at 17:18