If you’re interested in game theory, one of the most important concepts to understand is the calculation of game theory value. Game theory value is a way to determine the expected payoff of a game, and it can be used to help players make strategic decisions.
To calculate game theory value, there are a few key steps that you need to follow:
Step 1: Create a Game Matrix
The first step in calculating game theory value is to create a game matrix. This is a grid that shows all of the possible outcomes of the game, based on the choices made by each player.
For example, imagine that you’re playing a game with two players: Alice and Bob. Each player can choose between two options: “A” or “B”. The possible outcomes of the game are shown in the following matrix:
| Alice chooses A | Alice chooses B | |
| Bob chooses A | (3,3) | (0,4) |
| Bob chooses B | (4,0) | (1,1) |
In this matrix, each cell represents one possible outcome of the game. The numbers inside each cell represent the payoffs for Alice and Bob, respectively. For example, if Alice chooses A and Bob chooses A as well (the top-left cell), they both receive a payoff of 3.
Step 2: Calculate Expected Payoffs
The next step in calculating game theory value is to calculate the expected payoffs for each player. This involves multiplying the payoff for each outcome by the probability that it will occur.
To calculate these probabilities, you need to make some assumptions about what each player knows about the other player’s strategy. This is where things can get a bit tricky, as there are many different ways to model this kind of strategic behavior.
One common approach is to assume that each player knows the payoffs for each possible outcome, and that they use this information to make rational decisions. Under this assumption, you can use a technique called “backward induction” to determine what each player should do.
Step 3: Apply Backward Induction
Backward induction involves working backwards from the end of the game to determine what each player should do at each step. In our example game, there are only two steps (since there are only two players), so we can start at the end and work backwards.
At the final step of the game, each player has two options: “A” or “B”. We can calculate the expected payoffs for each option using the following formula:
Expected payoff = (probability of outcome 1 * payoff for outcome 1) + (probability of outcome 2 * payoff for outcome 2) + ..
For example, if Alice chooses A and Bob chooses A as well (the top-left cell), they both receive a payoff of 3. The probability of this outcome occurring depends on what Bob does – if he chooses A as well, it will definitely occur; if he chooses B instead, it won’t occur at all.
So we need to consider both possibilities when calculating the expected payoff for Alice’s choice of A:
Expected payoff for Alice choosing A = (probability that Bob chooses A * payoff for Alice when Bob chooses A) + (probability that Bob chooses B * payoff for Alice when Bob chooses B)
If we assume that Bob is equally likely to choose A or B (since he doesn’t know what Alice will do), we get:
Expected payoff for Alice choosing A = (0.5 * 3) + (0.5 * 0) = 1.5
We can do the same calculation for Alice’s choice of B:
Expected payoff for Alice choosing B = (0.5 * 4) + (0.5 * 1) = 2.5
Since the expected payoff for choosing B is higher, backward induction tells us that Alice should choose B.
Step 4: Repeat for Each Step
Now that we’ve determined what Alice should do at the final step of the game, we can move on to the previous step and repeat the process. This time, however, we need to take into account what Alice will do at the final step.
For example, if Alice chooses B at the final step (as we just determined), then Bob’s expected payoffs are:
Expected payoff for Bob choosing A = (0.5 * 4) = 3.5
Expected payoff for Bob choosing B = (0.5 * 0) + (0.5 * 1) = 0.5
Since his expected payoff is higher if he chooses A, backward induction tells us that he should choose A.
Conclusion
By repeating this process for each step of the game, we can determine the optimal strategy for each player and calculate their game theory value – i.e., their expected payoff under these optimal strategies.
Of course, this is just a simple example of how to calculate game theory value – in reality, things can get much more complex! But by understanding these basic principles and techniques, you’ll be well on your way to becoming a game theory expert.