In the early days of its (modern) history game theory focused a lot on zero-sum games. These are games in which total payoffs always add to zero no matter what the outcome. So, in a two player setting - your gain is my loss and vice-versa. It was arguably natural for game theory to focus on zero-sum games because they represent the epitome of conflict. The main reason the focus fell on such games is, however, more one of convenience - zero-sum games have a solution.
This solution is captured by the minimax theorem and all that followed. Basically it amounts to saying that there is a unique way of playing a zero-sum game if all players want to maximize their payoff and are rational. Most games do not have a 'solution', because there are multiple Nash equilibria and so there is not an obvious correct way to play the game. In this sense zero-sum games are 'nice' or 'convenient'.
But does it make sense to behave according to the minimax theorem? The simple answer is no. This is because the theorem takes as given everyone is rational, and expects everyone to be rational. We know that in reality people are not rational, so why should you expect them to be. To illustrate the point consider a rock-scissors-paper game between Alice and Michael. The payoffs below are the payoffs of Alice.
The essence of the 'solution' for Alice is that her choice should not be predictable. And, in a sense, this seems hard to argue with. If Alice is predictable in say, choosing Rock then Michael can pick this up and choose Paper. He wins. So, the solution is for Alice to randomly choose what she does in each play of the game. If she chooses randomly then she is unpredictable by definition.
Randomization is good because it means Alice has a 50-50 chance of winning. But can Alice not do better? Seen in a different light randomization seems defeatist because it means Alice limits her ambitions to a 50-50 chance of winning. If she thinks she can see a pattern in Michael's behavior then should she not try and exploit that rather than continue to randomize? Yes.
In reality we know that people are very poor at producing random sequences. So, if you are playing rock-scissors-paper it is highly unlikely your opponents strategy will be completely random. That opens the door for you to do better than 50-50. Note, however, that this means you are not randomizing either. Zero-sum games are not so much, therefore, about how well a person can randomize but more how well they can spot patters in another's behavior.
Are there ever occasions where it makes sense to randomize? Taking a penalty kick in football, serving in tennis, playing poker? The answer seems no. Two absolute experts might just randomize and take their chances, consistent with the game theoretic 'solution'. But, in all likelihood your opponent will not be completely random and that means you shouldn't be either. You just need to be better at predicting your opponent than he is of predicting you.
For an interesting analysis of how this can battle of prediction can be modeled and analyzed see the recent paper by Dimitris Batzilis and co-authors in Games (MDPI). They use level-k theory to analyze choice in the rock-scissors-paper game. It is roughly the case that a player with a higher level of reasoning will win. And experience seems a key factor in level of reasoning.
This solution is captured by the minimax theorem and all that followed. Basically it amounts to saying that there is a unique way of playing a zero-sum game if all players want to maximize their payoff and are rational. Most games do not have a 'solution', because there are multiple Nash equilibria and so there is not an obvious correct way to play the game. In this sense zero-sum games are 'nice' or 'convenient'.
But does it make sense to behave according to the minimax theorem? The simple answer is no. This is because the theorem takes as given everyone is rational, and expects everyone to be rational. We know that in reality people are not rational, so why should you expect them to be. To illustrate the point consider a rock-scissors-paper game between Alice and Michael. The payoffs below are the payoffs of Alice.
The essence of the 'solution' for Alice is that her choice should not be predictable. And, in a sense, this seems hard to argue with. If Alice is predictable in say, choosing Rock then Michael can pick this up and choose Paper. He wins. So, the solution is for Alice to randomly choose what she does in each play of the game. If she chooses randomly then she is unpredictable by definition.
Randomization is good because it means Alice has a 50-50 chance of winning. But can Alice not do better? Seen in a different light randomization seems defeatist because it means Alice limits her ambitions to a 50-50 chance of winning. If she thinks she can see a pattern in Michael's behavior then should she not try and exploit that rather than continue to randomize? Yes.
In reality we know that people are very poor at producing random sequences. So, if you are playing rock-scissors-paper it is highly unlikely your opponents strategy will be completely random. That opens the door for you to do better than 50-50. Note, however, that this means you are not randomizing either. Zero-sum games are not so much, therefore, about how well a person can randomize but more how well they can spot patters in another's behavior.
Are there ever occasions where it makes sense to randomize? Taking a penalty kick in football, serving in tennis, playing poker? The answer seems no. Two absolute experts might just randomize and take their chances, consistent with the game theoretic 'solution'. But, in all likelihood your opponent will not be completely random and that means you shouldn't be either. You just need to be better at predicting your opponent than he is of predicting you.
For an interesting analysis of how this can battle of prediction can be modeled and analyzed see the recent paper by Dimitris Batzilis and co-authors in Games (MDPI). They use level-k theory to analyze choice in the rock-scissors-paper game. It is roughly the case that a player with a higher level of reasoning will win. And experience seems a key factor in level of reasoning.
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