**Igal Milchtaich is from the Department of Economics at Bar-Ilan University in Israel. Recently he visited the department and gave a very interesting talk on polyequilibria. Afterwards Andy Lewis-Pye was able to talk with him…**

#### You gave a very interesting talk on polyequilibria. Perhaps you could briefly summarise for the layman what this work is about? Could you say a little bit about the motivation: do you think there are gaps in the existing game theory literature which need filling?

The central solution concept in game theory is Nash equilibrium, which is required to provide a precise, complete prediction about the players’ behaviour in a game. Polyequilibrium legitimises looser predications, or polyequilibrium results, such as “the equilibrium price is higher than five” or “the outcome is socially efficient”. Technically, a polyequilibrium is a collection of strategy profiles that share the property in question. Put differently, strategy profiles that do not have the property (say, those representing a socially inefficient outcome) are excluded. The exclusion has to be justified in the following sense. Every excluded strategy for a player has an adequate substitute: a non-excluded strategy that does at least as well against all non-excluded strategy profiles. A relatively small polyequilibrium, which excludes many strategy profiles, provides a sharper prediction about the outcome of the game that a larger polyequilibrium does. However, both are legitimate polyequilibria. Thus, this solution concept reflects a somewhat different philosophy than that of Nash equilibrium, in that it is content with learning something interesting about the players’ equilibrium behaviour and does require that the strategy choices be completely pinned down.

#### Are there applications you have in mind for your work in polyequilibria outside pure mathematics?

Yes. Consider the following simple example of bilateral trade. A buyer offers a price for a particular item, which the seller can only accept or reject. Suppose, for simplicity, that the item has zero value for the seller. It is then a reasonable strategy for the seller to accept any price greater than zero. In fact, this is a dominant strategy. But it is not an equilibrium strategy, because the buyer does not have a best response to it: offering any positive price *p* is less profitable than offering, say, half that price. The polyequilibrium concept solves this conundrum in the obvious way: it makes “offering no more than *p*” a legitimate choice for the buyer, for any given *p*. This is not a strategy but a polystrategy: a collection of price offers. Together with the aforementioned seller’s strategy, it constitutes a polyequilibrium, at which the result that the item is sold at a price not exceeding *p* holds.

Another area where the idea that the players’ strategies may only be partially specified seems very natural is dynamic games. In such games, the notion of subgame perfection requires that players not only respond optimally to the other players’ actual moves but would also do so off-equilibrium, that is, as a response to all possible deviations of the others from their equilibrium strategies. This is a very sensible requirement, as it eliminates non-credible threats: those that a rational player would not actually carry out. But in a large game, it can be quite cumbersome to check that it holds, as all counter-threats need also be credible, and so on. With polyequilibrium, it may be sufficient to specify actions only at some of the game’s decision nodes, say, those at or close to the actual path. For example, there is no need to specify a response to an action that is unequivocally detrimental to the acting player. Not specifying a response means that none of the possible reactions is excluded. This natural choice for the responder is, again, one that the Nash equilibrium solution concept does not allow.

#### More generally speaking, what is the state of play at the moment, with regard to applications of game theory? How much symbiosis is there between the interests of pure mathematicians or computer scientists and the interests of economists here?

Game theory is more relevant now than it ever was. The move to online economic activity means that clever and sophisticated trading mechanism can be implemented with relative ease. Ad spaces, for example, can be sold in large auctions where the reaction times are measured in milliseconds and the rules can be as complicated as one wishes. The design of good, efficient such mechanisms is a challenge to game theorists and computer scientists, whose pursuits increasingly converge.

A nice example for this convergence is matching problems: matching pupils to schools, residents to hospitals, kidney donors to recipients, and so on. The matching algorithms need not only be reasonably easy to implement as code but also have to be incentive compatible. That is, participants in at least one side of the market should be assured that stating their true preferences always leads to the best outcome they can possible get. Finding such algorithms, or even figuring out whether they exist, can be a non-trivial game theoretic problem. The practical importance of these matching algorithms is immense.

#### … so presumably these are some of the areas you would encourage younger people starting out in game theory to focus on? Any general advice you would give to somebody starting a PhD? What would you do differently a second time around?

There’s nothing as exciting and satisfying as blazing one’s own trail. Ultimately, you go where your ideas take you. The more you hear and learn, the more you expose yourself to new research directions, the greater is the chance that you’ll come up with something new. So my advice would be to start with whatever you find most exciting, but then not to be afraid of making sharp turns, pursuing new ideas as they come along. Another advice would be to try, from time to time, to make contributions that go beyond the incremental, to write papers that other people might find an inspiration, papers that will result in follow-up work. Attaining technical proficiency and “mathematical maturity” are also important, so my advice to PhD students is to take as many advanced math courses as they can possibly bear.

#### To finish on a lighter note – what do you enjoy outside of work? Which book did you read last?

A lively book I just finished is The Life Project, by Helen Pearson. It tells the story of the British cohort studies, which started in 1946. It’s a tale of science, scientific endeavour, and the sociology and politics of science. The heroes are the doctors and scientists who envisioned these studies and struggled to make them a reality and maintain them throughout the many, sometimes difficult years. Their achievements are the medical, sociological and economic insights that resulted from following the lives of the thousands of individuals involved and observing how their starting points and their decisions through the years affected their health, wealth and well-being, and the policy changes that this understanding helped bring about. It’s a story about the value and beauty of science – recommend reading!