# Stag Hare inefficient Nash Equilibrium and level-K thinking

Two players $$i,j$$; both have two strategies $$\{h,s\}$$.

The payoffs vector of $$i,j$$:

$$u(h,h)=(5,5)$$ (if both players choose $$\{h\}$$ then $$i$$ receives 5 and $$j$$ receives 5)

$$u(h,s)=(10,0)$$

$$u(s,h)=(0,10)$$

$$u(s,s)=(15,15)$$

The Nash equilibria include $$(s,s)$$ and $$(h,h)$$. Experimental evidence suggests that with communication, most people will realize $$(s,s)$$.

However, when information is incomplete, it could be rational for one player to play $$h$$ if one of the following conditions hold:

1. The player believes that hare is attractive enough for the other (this is discussed in a textbook); or, more generally if the player is a level-k thinker (k>0)
2. If the player is a max-min decision maker.
3. If the player is picking a Nash Equilibrium such that that his profit cannot be sabotaged by the other.

I wonder if the level-k results and/or the third interpretation ("non-sabotagable" NE) is reasonable or covered by literature.

Level-k reasoning in the stag hunt game is analyzed in

Gracia-Lázaro, Carlos, Luis Mario Floría, and Yamir Moreno. "Cognitive hierarchy theory and two-person games." Games 8.1 (2017): 1.

The idea that playing $$s$$ guarantees its payoff is discussed in

Aumann, Robert "Nash equilibria are not self-enforcing, in ‘‘Economic Decision-Making: Games, Econometrics and Optimization’’(JJ Gabszewicz, J.-F. Richard, and LA Wolsey, Eds.)." (1990).

The stag hunt game is due to Aumann.

Related to that is the idea of risk dominance of Harsanyi and Selten in their 1988 book "A general theory of equilibrium selection in games." The stag hunt is pretty much the leading example. It formulates the idea that this Pareto inefficient equilibrium is less risky to play.

It should be noted that the game has a third equilibrium in mixed strategies.

• Thank you so much! I cannot image that I did not find the Games 2017 paper on google scholar. Any tricks for literature search? Jul 3 '20 at 21:27
• I think the trick here was just looking for "stag hunt" instead of "stag hare." Jul 3 '20 at 22:01
• No, the stag hunt game is not "due to Aumann." The basic game was well known by the early 60s. The strategic problem was recognized much earlier. After all, the name comes from Rousseau, who laid out the basic logic in his fable of the stag hunter in 1755. And Hume discussed the problem a few years later. Oct 14 '20 at 23:15

As Michael Greinecker noted, the stag hunt is the leading example of a symmetric 2x2-game with a payoff-dominated but risk-dominant NE. In symmetric 2x2 coordination games, a pure NE is risk dominant iff it is the unique best reply to the mixture $$(\frac12,\frac12)$$. Since Level-0 types are usually assumed to mix uniformly over pure strategies, all higher-level types play the risk-dominant NE. In the stag hunt game this is hare.

Unfortunately, the numerical values you provide are such that both stag and hare are risk equivalent since both are best replies to $$(\frac12,\frac12)$$. To get a genuine stag hunt game, better choose some payoff $$>5$$ for both playing hare.

I'm not sure what a "non-sabotagable" NE is. If you mean one where your payoff cannot decrease if your opponent deviates, then this would coincide with the maxmin choice.

• It does coincide with maxmin in this specific example but not always, I guess. Jul 2 '20 at 17:46