There might be a hedgehog dilemma in a broad class
There might be a hedgehog dilemma in a broad class of games, but this paper focuses on the case where players only care about : the distance between their attitudes. Here we generalize the main example of a quadratic loss function.
So, player ’s payoff is higher when the distance is closer to ’s optimal point. The case of a quadratic loss function is the special case where . Without loss of generality, . Thus, ’s optimal distance is greater or equal than ’s. Player is called the remote and player the attached. The main technical challenge is the characterization of mixed strategy equilibria. This can be difficult even in the relatively simple setting where Assumption 1 holds.
The equilibrium where the remote pins down the attached does not depend on the optimal distance of the attached and the loss functions and . Within this model, , and can vary with no effect on this equilibrium. So, this equilibrium is robust and one-sided. The resulting distance is the best possible for the remote, but exacted on the attached, regardless of the attached preferences. There is no pure strategy equilibria (i.e., for , for some ) when optimal distances differ (i.e., ). Still, if there are multiple equilibria. Any player can pin down the other one in equilibrium. So, the attached can pin down the remote and obtain an ideal -distance. There are other equilibria as well. If then it dofetilide is an equilibrium for the attached to choose and with equal odds and the remote to choose and each with probability , and with probability , where In this equilibrium, both players randomize and, therefore, no player successfully pins down the other. The end result is inefficiency. This equilibrium is Pareto dominated by the one where the attached pins down the remote, but not, if is sufficiently close to , by the equilibrium where the remote pins down the attached. As approaches from above, both and the expected payoff of the attached go to zero. So, this equilibrium converges to the one in which the attached pins down the remote. This is a near perfect outcome for the attached. The main result of this paper is as follows:
If then players stay at the remote’s optimal distance . The greater the difference between players’ optimal distances, , the worse-off the attached is. The attached cannot escape this disutility in any Nash equilibrium, no matter how harmful this may be. The equilibrium is inevitably unequal. The attached is powerless to obtain anything, except for what is ideal to the remote. At , there is a jump discontinuity on the prospects of the attached. When is above the attached can stay at the attached’s optimal distance , but once is below there is only one equilibrium and it is uncompromisingly one-sided in favor of the remote. Above the threshold it is like, though not the same, a battle of the sexes, but below it the equilibrium is unique and disadvantageous to the attached. Finally, the threshold is independent of the loss functions. The loss functions and can differ significantly from each other, but no matter what they are, within the constraints of the model, the remote pins down the attached when . These results can be summarized as follows: For , let be the highest expected payoff that player can obtain in equilibrium. That is, for some equilibrium , and for all equilibrium , . Then,
The proof of Proposition 2 is involved, in part because when both players’ expected payoffs are not differentiable at critical points. The intuitive idea is to start with an equilibrium and then show that it has properties akin to the ones of equilibrium where the remote pins down the attached, until these properties become strong enough for the conclusion that it is that equilibrium. So, the proof starts with general results regarding continuity, differentiability and convexity of expected payoffs in some points (Lemma 1, Lemma 2, Lemma 3, Lemma 4) and then shows that outside the support of , , player , , has at most two optimal points (Corollary 2). After a few intermediary results, it shows that if the support of , , is sufficiently narrow then player , , optimal points are outside the support of , and conversely if the support of , , is sufficiently wide then player , , optimal points are inside the support of