Quantum Game Theory

28 December 2016
Posted by Frank Witte.

“Quantum game theory” originated in the late 1990’s as a branch of ... well let’s admit it ... “recreational physics”. But it has developed into an area that one should consider taking seriously! It is one of the currents in the growing stream of “Econophysics” literature that received a tremendous boost in 2008 after the near collapse of the global financial systems. So what is quantum game theory? And how does it relate to the “classical” game theory used in Economics and several Social Sciences to analyze decision making processes?

Classical game theory

Consider two “agents”, say Alice and Bob, who are in a position where they need to make a decision of some kind. They both have a set of strategies available to them and they both have a so-called “pay-off matrix” that tells them how much pay-off they will get from playing a particular strategy given that the other player plays another given strategy. The idea is roughly that both Alice and Bob choose their strategies, communicate that to the referee and then a referee determines the pay-offs. So, essentially, all the information available about the game is in the “pay-off matrix” and possibly in the order in which Alice and Bob choose their strategies. The goal of game theory is to analyse which strategies will actually be played by players. Such a set of strategies that will be played is commonly known as a “solution” to the game.

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Men arm-wrestling

A famous example of such a solution is the “Nash equilibrium” concept which states that “an equilibrium is a set of strategies where, if all other players’ strategies are fixed, a player does not have an incentive to change his strategy”. So one thing one can do in analysing a game is to use the pay-off matrices of the players to examine whether there is a Nash equilibrium. Nash received the Nobel prize in Economics for his work in Game Theory after years of struggling in a personal battle with mental ailments.

Evidently things become hairy when there is no equilibrium strategy or when there are several. It also happens frequently that the “solution” of the game happens to be a set of strategies that we all consider very undesirable as an outcome. An example of such a case is the “Prisoner’s dilemma” where two rational players are most likely to end up playing a set of strategies that is not optimal for either of them. Another example is the “Ultimatum game”. In this game Alice receives a certain amount of money and is asked to split it between her and Bob. If Bob accepts the split, each can keep their share. But if Bob chooses instead not to accept the split, both get nothing. Classical economic reasoning suggests that for Bob anything is always better than nothing, so Bob will accept any split. Knowing this Alice will offer him the smallest split possible. This doesn’t make Alice look very nice does it?

The ultimatum game has been experimentally tested in many different ways and the surprising result is that typical "Bobs" do not accept every offer. Also, typical "Alices" seem to know this, and making 30% offers is not uncommon. There is even an indication that this fraction is independent of the total stake. Classical game theory offers a very appealing way to analyse decision-making problems, but it has its difficulties in dealing with situations where there seems to be no or several solutions, as well as with situations in which measures of “fairness” play a role.

Let the Quantum Games begin

One way of trying to deal with these shortcoming is to remove the requirement that a “solution” must be played. Possibly the “solution” is merely the most likely strategy to be played? Introducing probabilities and statistics into game theory is standard practice in Economics, for example, and does not require any quantum stuff. Some games without Nash equilibrium do indeed show one as soon as probabilistic strategies are allowed. But others don’t. It is also difficult to see how probability could come into actual game play when a game is played only once? And it is at this point that quantum game theory has something to say.

Normative quantum games

Quantum game theory originated from the idea that the players could use “quantum objects” to communicate their strategy choice to the referee. For example Alice giving the referee an “electron” with either spin up or down could communicate a choice between strategy 1 and 2. The nice thing about electron spins is that they are quantum objects and can also exhibit super positions of these two states. This would introduce a fundamental uncertainty into the strategic choice of Alice. However this effect relies on the referee forcing Alice and Bob to use quantum objects. It is an example of a “normative description” of a decision-making process: we’re establishing a rule (“communicate through quantum objects”) and analyse the effects of that.

Quantum games offer a huge variety of extensions of traditional games and many instances are known where they indeed “solve” some of the conceptual difficulties with classical games. The offer all kinds of new effects such as “entanglement” that shines a new light on the phenomenon of “cooperation” between players. But the quantum character is enforced by requiring the players to use quantum objects for communication. Something which is technologically impossible at the present time. But it is also questionable why players would be incentivized to do so? Is quantum game theory indeed just “recreational physics” for science fiction authors?

Positive quantum games

A very different angle, which preserves all of the benefits, is to consider quantum games as an actual description of the decision-making process itself! This is a radical thought because it does away with the “rational player” entirely. Instead, the idea is that “decision-making” is very much a quantum-measurement process. From that point of view the players do not have a particular choice in mind until the referee forces them to choose. The choice that they then make is intrinsically probabilistic, and the probabilities for choosing one strategy over the other are determined by the “state of mind” the player is in.

The narrative of such an approach is very different from that of standard game theory. Players are no longer rationally calculating agents that arrive at some optimal conclusion. Instead they “live” a superposition of choices until a referee forces them to “make up their mind” the actual outcome of which is fundamentally unpredictable. To me this sounds perfectly reasonable for this happens to me all the time when I go shopping anyway. All the “quantum” aspects are now aspects fo the players rather than of the objects they play with. This version of quantum game theory basically claims that this is how we unavoidably work, rather than that players would need to be forced to work like this.

Outlook:

Economists generally are not convinced that all of this is any more than just physics play. However, after the financial markets disaster of 2008 people are looking everywhere for alternatives to current economic models and ways of thinking. Quantum game theory provides one such approach to bring some fresh air into the discussion on exactly how we make up our minds about things of value!

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