In 1994 the Bank of Sweden Prize in economic sciences in memory of Alfred Nobel was awarded to John Harsanyi, to John Nash, and to Reinhard Selten, for their pioneering analysis of equilibria in the theory of non-co-operative games. This theory includes situations such as the competitive tender.

Subsequently, John Nash found himself the focus of some media interest. John Nash’s biography became a best-seller in 1998 and a film based on the book A beautiful mind has won four Oscars. The popular interest was largely due to his tragic personal circumstances. In 1958, on the threshold of his career, paranoid schizophrenia struck. Nash lost his job at MIT in 1959 and was virtually incapacitated by the disease for the next three decades. He then staged a remarkable recovery, and now works once more at Princeton University.

Here is an example of the kind of problem Nash considered. Three firms are invited to submit sealed bids to make a widget. Each firm has a different cost of production, with the first costing $1, the second $2, and the third $3. They each submit a bid, and the contract is awarded to the lowest bidder. You are advising player 2 how much do you suggest they should bid?

What is a Nash equilibrium?

A Nash equilibrium describes a set of strategies for players in a non-co-operative game. The game’s equilibrium holds when each player’s strategy is a pay-off-maximising response to the strategies pursued by the other players. This concept has been the point of departure for most economic work in the field of game theory.

Nash’s key contribution was a mathematical proof that, for finite games, a Nash equilibrium exists. The proof itself is only a few pages long, but it relies on a deep topological result the fixed-point theorem proved by the Japanese mathematician Kakutani. There are a number of complexities to applying this in practice, including the problem of multiple equilibria and the strong economic assumption that a game’s participants are aware of each other’s constraints and preferences. The other Nobel winners have contributed to resolving these issues.

Mixed strategies

A player follows a mixed strategy if a random selection is made between various alternatives. Nash equilibria may involve mixed strategies, even when no deterministic equilibrium exists.

Consider our auction example. Player 2 will only rationally bid more than the production cost of $2, and player 3 will only rationally bid more than $3. If player 1 was aware of the other players’ strategies, they could immediately win the auction and make a profit of at least $1 by bidding one cent below the cheaper of the other two. From player 1’s perspective, this is plainly an optimal strategy.

To confound this activity, players 2 and 3 could choose their strategies randomly. In other words, each could choose their bid from a suitable probability distribution. The purpose of these random choices is not directly to maximise their own profit, but instead to confound player 1’s effort to undercut. By making random bids, players 2 and 3 gain the chance that player 1 will be too greedy, enabling either players 2 or 3 to undercut while still taking a profit.

The auction analysis

Let us now return to our auction example and examine a Nash equilibrium. Each supplier chooses a random bid independently of the other players. All bids are in excess of $5; the amount of the first player’s bid is a random sample from a distribution with cumulative distribution F1(x), where

x<=5

F1(x)=P{bid<=x}={1? x>=5

The other players follow similar strategies, with distribution functions as follows (these apply on x>=5, the functions are zero below this):

F2(x)=1?

F3(x)=1?

From charts of these curves (see figure 1), we can see that player 3, with the higher unit cost, is likely to quote a larger bid. This is not necessarily a losing bid, as all players select their bids randomly, so even with a very large bid player 3 may get lucky.

How can we verify that these strategies together constitute an equilibrium? We need to demonstrate that no player can improve their expected pay-off given the strategies of the other two players.

So let us look at player 2’s strategy. If x is played, then there is a probability of [1F1(x)”[1F3(x)” that the other players both bid a higher value. In this case, the gain to player 2 is x2, otherwise the gain is zero. The expected gain to player 2 is then

[x2”[1F1(x)”[1F3(x)”

= (x2).? .?

= 3

In other words, given the strategies of players 1 and 3, player 2 is indifferent between all bids x>=5, and in particular, the equilibrium mixed strategy is no worse than the alternative deterministic ones. Indeed, this is how I derived the distribution function. Players 1 and 3 have done a great job in confounding player 2, neutralising any effort on player 2’s part to exploit known competitor strategies. Similar results apply from the perspectives of each of the other players.

This is not, however, the unique Nash equilibrium for this problem. You could vary the solution by looking at other minimum bids, higher or lower than the $5 I have shown here. There is a much simpler equilibrium where player 1 bids $2, and the other players bid marginally above that. Game theory tells us little about which of these equilibria we should expect to see in practice.

Does it work?

The relevance of Nash equilibria to modern economic practice is still controversial. Perhaps the most promising area is the design of auctions. These auctions can be long and protracted affairs. In April 2000, the UK government announced the results of its own airwave auction: after 150 rounds, the accepted bids totalled £22,470m.

This proved to be the high point of the telecom auction frenzy. Does game theory deserve the credit for Gordon Brown’s windfall? Or were the fluctuations simply reflecting the boom and subsequent collapse in the fortunes of telecom companies?

I have carried out some tests of a simple auction game, both within my own office and in a workshop at the general insurance convention. I chose a problem with a unique Nash equilibrium, in which each player has a deterministic strategy. In the office, my colleagues quickly converged to Nash equilibrium behaviour, while general insurance volunteers probed a wider range of strategies. By the time you read this article, life actuaries will also have had the chance to try this game.

Much as I would like it to be true, I would hesitate to conclude from this that my colleagues are smarter than other actuaries. Instead, these experiments show that Nash equilibria for simple games are not necessarily predictive of actual outcomes. This could be because the players fail to conform to game-theoretical definitions of rationality. Equally well, it is possible that my formulation of the game has failed to capture actual players’ real objectives. The workshop had a Have I Got News for You air about it players may have tried harder to entertain the audience than to win the game. In our auction example, there may well be a price above which the auctioneer will abort, choosing instead to negotiate privately with player 1 in which case different optimal bidding strategies emerge.

The difficulty of formulating even simple problems in a game-theoretic framework is serious. There are many parameters to estimate to formulate the problem, most of which relate to hypothetical pay-offs under strategies, which have not in the past been followed so for which no supporting data is available. Controlling the external incentives in laboratory type tests is close to impossible. On top of this, Nash’s equilibrium theorem is merely an existence result, which gives us no guidance on how to characterise numerically the set of Nash equilibria.

Actuarial applications?

Are Nash equilibria ever going to be of practical use to actuaries? Is this a fast developing area where actuaries must work hard to catch up, or is game theory a peripheral discipline which most of us can leave to the specialists?

In 1959 the late UK actuary Sidney Benjamin published one of the few attempts to apply this literature in an actuarial context in this case to a minimax definition of prudent valuation bases. Others have claimed to apply game theory to life insurance underwriting, market volatility forecasting, timing of market sales and purchases, capital allocation, insurance premium cycles, and even the management of terrorism risk.

In all of this, I am not aware of a single example where a manager has sought to calibrate his own and competitors’ pay-offs and then successfully forecasted the future by solving for a Nash equilibrium. Instead, most claimed applications of game theory involve ideas, concepts or insights used in a judgmental fashion. A cynic might question whether game theory is actually being applied at all. Claims to use Nash equilibria may turn out to be the structured application of general reasoning, given a veneer of sophistication by the adoption of a Nobel-winning name.

Game theory is still developing rapidly. It holds out a promise as yet unfulfilled of explaining puzzling effects in insurance and capital markets. Actuaries should keep abreast of developments, and be ready to adopt game theoretic tools as they become more practical to apply.

02_12_09.pdf