## Zero-sum games

Game theory is the mathematical description of the structure of strategic decisions when the choices of two agents directly influence the other. Much work has been dedicated towards finding optimal strategies; in fact much military and political strategy is built on mathematical foundations, but it also governs economic decisions.

Consequently some knowledge of game theory can be very helpful.

The simplest way to describe a game is by using a payoff matrix. Suppose player 1 has strategies A and B and player two has strategies C and D. Then there are 4 outcomes: AC, AD, BC, and BD each with a certain pay out.

The most famous game theoretic problem is the prisoners’ dilemma, where two criminals are interrogated by the police. In the prisoners’ dilemma, a prisoner has the choice of ratting on his partner or keeping quiet. Here is the payoff matrix. Hence quiet-quiet = 3 years for both, quiet-rat = 10 years and 0 years. rat-quiet = 0 years and 10 years and rat-rat = both 5 years.

What is the optimal strategy? It turns out that the optimal strategy for the individual is to rat on your partner. Ratting on your partner is a so-called Nash equilibrium after John Nash who was portrayed in the movie A Beautiful Mind. A Nash equilibrium is the optimal strategy for an individual given full knowledge of the other player’s strategy. In other words it is the choice of strategy where the payout of the individual can not be improved by changing strategy. In the case of the prisoner, since 0 years is better than 3 years and 5 years is better than 10 years, a prisoner can not improve his position by keeping quiet

This is an interesting case where the global optimum will not be chosen. This is what makes it a dilemma. People with too much sparetime can now make a game theoretic model of whether for two hostile nations it is in the best interest to for a non-nuclear nation to pursue a nuclear program given the idea of a Nash equilibrium (answer: it is).

Now, bripblap asked if the same holds in terms of wealth.

Let’s look at that. I’d like to rephrase the question in terms of whether trade has a global optimum (it does). When trading people are often confused by vague definitions of price and value, so let’s define those first. Price is the objective measure of at which a trade took place. If I traded 10 hogs for 10 cows, the price of 10 hogs is 10 cows and vice a versa. Value is wholly subjective. I am trading my 10 hogs for 10 cows because I believe 10 cows are of greater value than 10 hogs. The person I am trading with believes the opposite.

Drawing a payoff diagram in terms of value shows that the optimal strategy in a free market is to only accept a trade when I believe that I am receiving more value (wealth) than I am delivering. The trade does not take place unless both counter-parties think the same. Hence, value is always created. That is the beauty of a free market system.

Conversely, if the trade was coerced (taxation), at least one party would think he is getting less value than he is receiving. This can lead to zero-sum or even negative-sum games. So why do we do this? The reason is that payoff is not strictly linear. Most humans, for empathetic reasons believe that a situation where a majority of people are slightly discomforted is optimal if it means that one person avoids severe pain (disutility). In other words, when it comes to extreme hardship, the needs of the one outweighs the needs of the many.

Another thing that can lead to negative payoff is buyer’s remorse. Although a rational person would base decisions on all available information at the time of the trade, few people are rational. Hence people will often choose suboptimal strategies. This, as well, can lead to zero or negative sum games. This does, however, not necessarily mean that the other person gets more value, but it is often so.

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Originally posted 2008-11-07 07:36:27.