The short answer is: no.

That is a watered down version of the conclusions in the linked paper by Matthew O. Jackson, Tomas Rodriguez-Barraquer and Xu Tan.

The link is to an ungated version, but the final version of the paper can be found in Games and Economic Behaviour. The title of this post will probably require a bit of background for some readers, so here goes.

For non-economists:

One of the most fun parts of economics is game theory (it is totally as fun as it sounds!). Game theorists use mathematical tools to analyse behaviour and find equilibria in strategic situations. We can somewhat informally think of these equilibria as predictions; in an equilibrium everyone is happy with their actions (given everything else that is happening in the game) so no one wants to change their behaviour.

A key problem that economists have been working on since at least the 1970s is the fact that most games have multiple equilibria. This is a real problem when you are trying to make predictions – which of the equilibria should you choose as your prediction for the game? Despite working on this for 40 years, economists are yet to come up with an answer that can be applied universally to all games. In fact, economists now think that this question is pretty much impossible to answer – if you give me an algorithm for picking an equilibrium to use as your prediction, I will be able to find a game where your prediction is either absurd or not unique.

The other thing we need to know about is epsilon equilibrium. An epsilon equilibrium is a type of equilibrium that comes about when people don’t care about very small changes in their payoffs. Ask yourself: if someone gave you 5 cents would you feel any better off, even a tiny bit, than you were before? What about 1 cent? What about half a cent? What about one hundredth of a cent? If you answered ‘no’ to any of these questions, then you are a candidate for epsilon equilibrium.

Jackson, Rodriguez-Barraquer and Tan ask the question: can we use the ideas behind epsilon equilibrium to help us choose an equilibrium to use as our prediction? And their answer is no, it is completely useless. Not only can it not make a unique prediction it can’t even rule out any equilibrium, ever.

But now comes the really depressing part. Over the years, economists have come up with a lot equilibrium refinements that work in a reasonably broad range of cases. J, R-B and T use their results to show that a whole lot of these refinements (even some pretty popular ones) can only work if we are absolutely sure that we know everyone’s payoffs in the game absolutely exactly.

If we are even a little bit uncertain about the payoffs in the game (i.e. we are not entirely sure that you like outcome A exactly 2.06 times more than outcome B) then we are in a situation where we can use epsilon equilibrium to approximate the game in question. And when we use epsilon equilibrium we can’t rule out any equilibrium, ever. So none of the equilibrium refinements that rely on approximations to the true game can ever work, unless we are absolutely sure that we have the payoffs correct!

In a sense, you can think of this paper as being a big middle finger to the thousands of papers on equilibrium refinements that have been published over the last 40 years. But in a more positive light, we did learn something from Jackson, Rodriguez-Barraquer and Tan: if you think that you might have misspecified the payoffs in your game, it is important to make sure that you use techniques are robust to the potential misspecification. J, R-B and T have shown that a bunch of techniques that we thought were robust are actually not. This is progress.

For economists:

The main result that Jackson, Rodriguez-Barraquer and Tan prove, as stated in the abstract, is that for any equilibrium of a Bayesian game, and any sequence of (payoff) perturbations of that game, there exists a corresponding sequence of ex-ante epsilon equilibrium that converges to the equilibrium of the original game.

The key implication of this is that any equilibrium refinement that relies on perturbations to some aspect of the equilibrium will break down if we also allow for perturbations in the payoffs of the game. The most obvious application of this result is to trembling-hand perfect equilibrium: if we allow for perturbations in payoffs as well as mistakes in play then we can longer use trembling hand perfection to rule out any equilibrium.

Note the use of ex-ante epsilon equilibrium. The definition of ex-ante epsilon equilibrium requires that, for any strategy used in the epsilon equilibrium, the ex-ante payoff (i.e. before private information regarding types is revealed) for the strategy is within epsilon of the ex-ante payoff for the optimal strategy. This leaves open the possibility that behaviour can be far from optimal for some type realizations (as long as these realizations occur with a small enough and vanishing probability).

The result can be strengthened to the notion of interim epsilon equilibrium if beliefs in the perturbed games converge strongly enough to the beliefs in the original game. Interim epsilon equilibrium requires that almost all types earn an interim payoff (i.e. after private information is revealed) that is within epsilon of their best response payoff.

Jackson, Rodriguez-Barraquer and Tan also demonstrate, in a remark hidden towards the end of the paper, that any sequence of epsilon equilibria that converges to a strategy in the original game will be  Bayesian equilibrium of the original game. I think this result should be more prominently placed in the paper; although it is less surprising than the key results of the paper it provides important context on the properties of sequences of epsilon equilibria that readers not heavily invested in the literature may not be aware of.

While this paper can be viewed as another blow for the refinement literature, there are some positives. As Ryan Oprea pointed out to me, it implies a more important role for experimental and empirical evidence in the task of equilibrium selection.