The horizon between profit maximizers and loss minimizers
Minimax Theorem in Game Theory
In Game Theory, there are many different methods a player can use to maximize their probability of winning a game. In theory, we usually take N rational thinkers and map out all the possibilities. These rational thinkers always take the best decision to improve their situation. But in real life, different people have different approach methods for the same problem, people are born with different lenses. Some are profit maximizers, while some are loss minimizers and the rest are somewhere in between.
Let us take the example of a simple Stag-and-Rabbit hunt game. There are 2 wolves in the wild, hunting for meat. They both have 2 options, to collaborate and hunt a bigger animal, like a stag, or to go off on their own and hunt a smaller animal, like a rabbit. In collaboration, there is an ambiguous factor of relying on the other wolf for teamwork, and not following through will result in a Tuesday for both the wolves. The payoff table will look like this:
We can see that there are 2 points of Nash Equilibrium in the payoff table, one at (2,2) and the other at (1,1). But if we play out this scenario many times, then we will see both the wolves leaning towards teamwork (2,2) because it is the better choice of both the equilibrium points.
The term minimax is a combination of the words minimum and maximum, in this strategy a player chooses the highest value (maximum) of all the worst possible scenarios (minimums) to completely avoid the worst payoffs.
Now, to bring minimax into play, let us alter the values of the payoff table by increasing the penalty from a mere 0 to -100.
Now once again, the 2 points of Nash Equilibrium remain the same but the way the players(in this case wolves) approach the problem will change. Both the players are still aware that (2,2) is the better option but their payoff also depends on the decision of the other player and if one player relies on an individual hunt then the other player will be in a really bad spot.
Increasing the penalty increases the chances of ending up at the bottom right of the payoff table even after playing out the scenario many times, especially when some people can make erratic decisions and at times be random.
In fact, many people will consider what is the worst possible scenario here and try to completely avoid it.
Now let’s see an example where the players would have more than just 2 options.
Let us take the scenario of a war struck city where the war has been going for some days now. There is a mansion which has been infiltrated by soldiers of both sides. Both the soldiers are aware of each other’s presence in the mansion but don’t know the other soldier’s exact location. They will have 5 choices, running, hiding, fighting, playing dead, or freezing.
Below is a payoff table for 2 soldiers (or players).
This time we have also added a row and column to display the worst possible payoffs (or minimums) for each strategy that the player uses. If player 1 chooses to play dead, then the worst payoff he will have out of the 5 possibilities is a 2, the same goes for player 2 when he chooses to play dead.
If a player uses the minimax theorem to make his decisions, then he will choose the maximum payoff of those minimums. So for player 1, the maximum of the possible minimums (4,1,2,0,0) is 4 points, which means choosing to run. Applying the same theorem for player 2, the minimax payoff will be choosing to play dead which will help him get a payoff of 2 points.
Thus, the minimax equilibrium will be established at (5,5), where player 1 chooses to run and player 2 chooses to play dead, giving them both a payoff of 5 points.
Please note that the minimax equilibrium being established at (5,5) is just a co-incidence, the parties involved will not always have equal payoffs in every scenario.
As previously mentioned, everyone has a different approach for the same problem. Some people will consider not having a negative payoff as a positive payoff. If they get a payoff of 0 instead of -5 then they will see it as a gain of 5 points, and there’s nothing wrong with that. In fact, having such an approach will help the player ensure that he is not at the bottom of the scoreboard in a game played by more than 2 players but it’s hard to determine whether such an approach will ensure that the player will be at the top of the scoreboard.
So now that you know what minimax is, are you a loss minimizer or a profit maximizer?
