Professor Andrew McLennan is an internationally renowned researcher in the fields of mathematical economics and game theory. In this Q&A, he provides some insight into the world of game theory, his current and future projects, and the value of his research.
1. What is game theory?
Game theory studies mathematical models of social interactions, or “games”. Game theorists analyse the strategic decisions made by people when they compete or cooperate with others.
A key concept in game theory is an equilibrium – a strategy for each player that is best for them, assuming the strategies of the other players are fixed. Game theorists can ask whether the actual behaviour of the players is accurately described by the equilibrium, and if not, why. They can also think about changing the rules of the game to improve the equilibrium outcomes.
2. Can you give me a real-life example of game theory in action (something students would understand)?
An auction is a real-life example of a game. Auctions are increasingly important on the internet and for problems such as privatisation and allocating rights to exploit mineral resources on public land.
If you are bidding on a house, how much should you bid? Certainly, you should bid less than the most you would be willing to pay, but the lower your bid, the smaller your probability of winning. If you are selling your house, would you be better off using an open outcry auction –used by auction houses like Christie’s and Sotheby’s – or should you follow the standard practice of asking people to submit sealed bids? Should you have a reserve price? Game theory helps us to understand why people do what they do, and whether they could do better.
3. What are you currently researching in the field?
Take dorm rooms: they can be allocated fairly by ordering the students randomly, then going down the list, with each student choosing their favourite room from those that have not already been claimed. In 1979, Hylland and Zeckhauser from Harvard University proposed a different model for problems of this sort, in which the players trade probabilities of receiving the various rooms in a market.
A key mathematical issue, both for game theory and for markets, is to prove that there is at least one equilibrium. My latest project gives an equilibrium existence theorem that simultaneously generalises their existence result and the most general results for standard markets. An existence result is a mathematical theorem that guarantees that a certain type of model has at least one equilibrium. Without an existence result, the story the model is trying to tell falls apart.
4. What is it about your research that is new or distinctive?
Much of the most interesting research in economics is basic, dealing with mathematical frameworks that are abstract representations of fundamental economic issues. My style of research combines a high level of mathematical knowledge with a sharp eye for researchable problems that are interesting, both mathematically and economically. My contributions are generally quite technical, and quite diverse, spanning a broad range of economic theory.
5. Which issues/research areas do you intend to tackle next?
Although the markets proposed by Hylland and Zeckhauser have several advantages, the possibility that there might be more than one equilibrium makes them hard to use and hard to justify in terms of fairness.
My research will provide quantitative evidence that multiple equilibria are common. I hope to provide a precise description of the set of equilibria when each person cares a great deal about the difference between their favourite and their second favourite outcome, a little about the difference between their second and third favourite outcome, only slightly about the difference between their third and fourth favourite outcome, and so forth. (For example, suppose your favourite outcome is worth $1 million to you, your second favourite is worth $10,000, your third is worth $100, your fourth is worth $1, and so on.)
