### Shannon Bohman

For half a century, game theory and quantum mechanics were two academic disciplines that were entirely dissociated from each other. Twenty years ago, however, scientists began to speculate that the same principles governing the tiniest particles in the universe may be useful in playing games. Today this unlikely union is a fruitful area of research known as quantum game theory.

Game theory is the branch of mathematics that studies how rational entities make decisions. The field had militaristic origins, but it has proven essential in understanding a wide variety of scenarios, from rock-paper-scissors to poker to the stock market. Game theorists are particularly interested in generating outcomes that allocate the most payoff to as many players as possible. When it is no longer possible to increase the payoff of any player without decreasing the payoff of another player, the outcome is said to have achieved Pareto efficiency. This is an important benchmark of which to be conscious, especially in economic planning and resource allocation. It is at this point that any further allocation necessarily harms at least one other party.

While game theory tracks the movement of entities as tangible as money and resources, quantum mechanics studies the behavior of the smallest particles known to physics. There are two key principles: Superposition is the notion that a physical state is the sum of multiple quantum states. Entanglement occurs when the quantum states of multiple particles are dependent on each other, no matter the distance between them. At first glance, these esoteric physical concepts seem disparate from game theory. Nevertheless, researchers have shown that superposition and entanglement can be useful to producing efficient game outcomes.

In “A Review of Quantum Games,” Gaon Kim of K. International School Tokyo and Eung-won Nho of Chungnam National University assess twenty years’ worth of literature to identify overarching conclusions and yet unresolved questions about quantum applications in game theory. They cover four categories of quantized games: quantum simultaneous non-zero-sum games, quantum simultaneous zero-sum games, quantum coalitional games, and quantum sequential games.

Simultaneous games are those where the players make decisions at the same time. In order to be played, there must be a common understanding among all players of the rules and possible options. The situation grows more complicated with added restraints on how players can earn winnings. In a zero-sum game, the net payoff of all players must balance the net loss. That means if one player wins more, at least one other player must win less such that the sum of all winnings exactly equal the sum of the losses. Conversely, net payoff and net losses need not be equal in a non-zero-sum game.

Simultaneous non-zero-sum games between two players may be quantized by superimposing each player’s strategies and entangling their decisions. Doing so produces a Pareto efficient outcome in which each player enjoys a higher payoff than possible from a classical solution. This solution, though, is somewhat perplexing. Entangling decisions leads to strategic coordination between players, which seems to counteract the simultaneous nature of the game. The current literature maintains that solving simultaneous games with strategic coordination does not in fact break any rules of the game, since the strategies each player employs are still chosen independently. However, the mechanisms by which entanglement generates this independent strategic coordination remain unknown. Such is a worthwhile area of future work. Simultaneous zero-sum games enjoy similar benefits through superposition of strategies without entanglement, with both players seeing higher payoffs.

Sequential games, such as chess, have the players choosing one after another, affording players the added information of their opponent’s previous turns. Quantization in this case is nearly exactly the same as it is in simultaneous games, the only difference being that entanglement and superposition are applied to a sequential structure. Coalitional games, sometimes called cooperative games, group the players into teams that compete with each other under an enforceable ruleset. Quantizing these games requires creating a shared entangled state among players and superimposing odd and even states separately. Quantum sequential and coalitional games also see improved payoffs due to strategic coordination, but as with other types of games, the precise mechanism by which these improvements occur is still unclear.

Each of the games discussed, though solved on a quantum computer considering only a minimum number of players, serve as a microcosm of important situations that arise in the real world. A football coach, a military general, and a social welfare manager may have nothing in common save for the fact that their success all hinders on being able to make strategic decisions. Employing game theory assists decision makers in making the best moves given a set of circumstances. Kim and Nho’s review emphasizes the consensus among game theorists that quantizing games enhances outcomes for everyone involved in the game. They also note the dearth of research that exists pertaining to the disparity between different quantization schemes and the role that strategic coordination plays in improving payoffs. But since the consequences of quantum game theory are so socially significant, one can expect that these questions will soon be answered.