Game Theory

This topic is one of the main parts of what are known as Decision Mathematics. Naturally, the title Game Theory makes the field sound like people playing chess or poker; doing recreational activities. This is actually not the case.

Game Theory is all about making choices, in given scenarios. What makes these scenarios special is that you are not the only one making choices; others are doing this as well and the outcome is affected by their choices as well as yours. One such example is that of conflict. In a battle between two opposing sides, each side may choose to fight or retreat (there are actually many more choices, but I am simplifying here for illustration purposes). If one side fights and the other retreats, the side that fought gains some advantage, e.g. some high ground. In game theory, we may assign a certain amount of points to this outcome, for example the attacking side gets 1 point, whilst the retreating side gets -1 point, to quantify the advantages gained and lost. On the other hand, if both sides fight, they will have a lot of casualties, so both sides get -2 points. Finally, if both sides retreat, they have not yielded anything to the opponent, so they both get 0 points.

Let us assume you are a military advisor of sorts, to one of the sides. What would be the best plan, for this battle? Should you attack or retreat? If we look at the possible results, fighting will get you either 1 or -2 points, whilst retreating will get you 0 or -1. If you want to make sure that you do no worse than a score of -1, you should retreat. This is known as the maximin strategy, as it seeks to maximise the minimum possible result. On the other hand, attacking ensures that the enemy will certainly have a score of at most -1. This would be an aggressive tactic. Of course, if the opponent is known to be aggressive, attacking will result in the worst possible score, for both sides. On the other hand, if the enemy is known to retreat a lot, attacking will cause a gain in points for your side.

As you can see, this sort of decision making is not always easy. In fact, depending on the scenario presented, it could be that every choice is a "bad" one. Or, alternatively, there can be an "obviously good" choice, which is always the best to choose. All these considerations are what Game Theorists study. From the above, it is obvious that the ideas of game theory have a clear practical application in such things as warfare (as well as politics, business management and so on). This may seem like a stretch, considering the simplicity of the scenario I presented, but the ideas were applied during the Cold War, amongst others. And since nuclear arms were already available by then, it can be seen how desirable it is that military advisors should be able to identify strategies such as "minimax" as the best ones to take.

As an exercise, you can try to teach yourself some elementary decision-making. Make your own scenario for two sides in a game, each having two options and each pair of options resulting in a certain point score. It should be fairly easy to figure out what scores make a perfect strategy possible. What about more choices? Or more players? Or both? Is it always clear if a scenario has a perfect strategy or not?