Game theory is a branch of mathematics with a lot of applications. Especially in economics, game theory plays a very important role. That’s the reason why a lot of mathematicians are presented with the Nobel prize in economics. A widely known example is John Forbes Nash, who’s life is illustrated in the movie “A beautiful mind”.
As the name “Game theory” suggests, there are some kind of games involved. In fact “game” is very broadly defined. It means a scenario where two (or more) perfectly rational “players” have to make some kind of decision. The decisions they make determine whether or not the players win or loose. But let’s see a game in action.
A simple game
There are two players, Alice and Bob as well as a game master. The game works the following. Both Alice and Bob have the choice of either giving 8 € or nothing to the game master. The decision whether to invest or not is made without knowing the decision of the other player. As the next step, the game master takes all the invested money, which is 0 € if Alice and Bob both gave nothing, 8 € if Alice or Bob invested but the other player refused or 16 € if Alice and Bob invested and raises it by 50 percent. All the money, including the interests of the game master, gets shared between the two players.
So let’s say both players invested the 8 €, which results in 16 € invested in total. The game master raises the money by 50 percent to 24 €. After equally sharing the money to Alice and Bob, both made a plus of 4 €. However having both Bob and Alice investing is only one case. It could also have been that either Alice or Bob or even both declined the investment. In mathematics we don’t like actually playing such games, we want to understand and solve them completely, so we never have look at them again. So let’s do this. In the table below all four possible outcomes are listed with corresponding profit or loss for Alice and Bob.
But hold on. Something in this illustration looks suspicious. Let’s say Bob invests. What’s the best strategy for Alice? Declining! Ok, then let’s say Bob declines. What’s now the best strategy for Alice? Still declining! However, also for Bob the best strategy is to always decline. So in the fictional world of maths, where both players act perfectly rational, Bob and Alice will never invest anything and never win anything, although they had the chance to both come out as winners.
There are plenty of real world scenarios which can at least be partly described with the game above. Just think about a team project in school, where students are not willing to invest their time out of the fear to do more work then others. If however all students would contribute to the project, they would profit from a great outcome.
Extending the game
Now, why is our game of interest at all? Is this the bitter end for Alice and Bob or is there a way they can – in the perfectly rational world – collaborate and make some money? Yes there is! The magic happens, if multiple rounds of the game are played. In that case the number of possible strategies explodes. Bob could e.g. always decline or always invest, but also more complex strategies are possible like investing every third time and declining every other time. Because Bob has a lot of strategies to choose from, Alice can’t for sure choose the most optimal strategy without knowing the one of Bob. As a result a lot of scenarios with sometimes complex patterns emerge.
The most important requirement for this game to work is that Bob and Alice don’t know when the last round is. Otherwise we’d be back at the game with only one round. Because e.g. let’s say we play ten rounds and both Alice and Bob know that. Obviously they will both decline in the tenth round for the reasons mentioned in the first game. Because the tenth round is essentially not existent, they’ll decline in the ninth round as well. That’s why it’s very important that the number of rounds played stays secret.
This game also happens in real life. It is a theoretical model of an economic interplay, namely the system of the job market in a social market economy where people with no jobs get an unemployment benefit. Each month a person can decide whether or not to invest their time. Because not knowing which is the last round is important for our game to work, in some sense, our economic system would crash if we all knew our future.
Whatever, back to game itself. The question for the best strategy is still open. There exists an infinite amount of strategies with any desired complexity. For example a strategy could be to invest only if the sum of the prime factors of the current round number is a power of three. But forget about that. The answer is surprisingly simple.
– In the first round invest
– In all other rounds do what the other player has done the round before
Why is this an optimal strategy? Let’s say Bob plays with this strategy. If Alice always declines, Bob only looses in the first round but is safe afterwards. If Alice always invests, then Bob also always invests and wins money. Assume Alice invests twenty rounds, then declines twenty rounds and so on. Bob will with his strategy almost adapt to strategy of Alice with the only difference to being one round off. Basically Alice can choose whatever strategy she wants, with our strategy Bob can loose no more than 8 €, but will win money if also Alice is willing to invest.