The Law of Total Probability states that the payoff for a strategy is the sum of the payoffs for each outcome multiplied by the probability of each outcome.
A simple example illustrates this law. Suppose there is an interaction in which you could either win or lose. There are two outcomes (win and loss), each with its own probability. According to the Law of Total Probability, the payoff is:
(probability of winning) × (payoff if you win) + (probability of losing) × (payoff if you lose)
To make this easier to write, we’ll represent the probability of an event as Pevent, so now we have:
Pwin × (payoff for win) + Plose × (payoff for loss)
How do we know the probability of each outcome? Since we want to find the average payoff for all players of the strategy, we imagine the probability for an average member of the population, that is, one who is of average size, fighting ability, and so on. In this simple example, that means that the probabilities of winning and losing are equal, at ½. (You could also reason that each interaction has a winner and a loser, so there are equal numbers of winners and losers in the population, making the probability of each outcome the same.)
When there are more than two possible outcomes, there are more terms in the sum:
POutcome 1 × (payoff for Outcome 1) + POutcome 2 × (payoff for Outcome 2) + … + POutcome N × (payoff for Outcome N)
For example, in a betting game that depends on the suit of a card that is drawn from a full deck, your payoff would be ¼(payoff for club) + ¼(payoff for spade) +¼(payoff for diamond) + ¼(payoff for heart).
Or, for a more complex example, consider a game in which you roll dice and you get one payoff if the number is 1-3, another if it is 4-5, and another if it is 6. Your payoff would be (1/2)×(payoff for 1-3) + (1/3)×(payoff for 4-5) + (1/6)×(payoff for 6).
You may have noticed that the probabilities add to 1 in all of these examples. This is no accident, and when calculating average payoffs, the probabilities must always add to 1.
Let’s take as an example animals fighting over a resource. For simplicity, we’ll say that the resource value is v and that the cost of losing a fight is c. Whenever two animals fight, there is a winner, who gets the resource, and a loser, who gets nothing and incurs a cost. The probability of winning and the probability of losing are equal, at ½. Thus half the population gets v and half gets −c. The payoff is:
Pwin × (payoff for win) + Plose × (payoff for loss), which is ½×v + ½×−c, or v/2−c/2.
How does it work in less extreme cases? After all, the winner may not get everything and the loser nothing. A winner might get most of the resource and the loser the rest, with the costs being similarly divided. Even when we consider these cases, however, the average outcome is still v/2−c/2. For every winner who gets ¾, a loser gets ¼, which averages to ½. If a winner gets 2/3, the loser gets 1/3, again averaging to ½, and so on for any other division of the resource and cost between winner and loser.
This illustrates that, when thinking about payoffs, we can usually simplify our reasoning and still get the right answer. In this case, we simplified things by making the outcome all-or-none, v or −c.
When filling out a payoff matrix, you need to do this calculation for each pair of strategies. Of course, the probabilities may differ depending on the strategies. The above example was for two animals using the same simple strategy, fighting. With other strategies, calculating the probabilities may be trickier.
Some strategies are conditional, in that the user of the strategy acts differently depending on circumstances. For example, “fight if I’m larger than my opponent, but back off if I’m smaller” or “fight to keep ownership of a resource, but don’t fight if someone else already owns it” are both conditional strategies. The action depends on a condition such as size or ownership.
The total payoff depends on the probability of each condition being met and on the outcome of each action. So if Condition 1 leads to Action 1, Condition 2 leads to Action 2, and so on, the total payoff is:
PCondition 1 × (payoff for Action 1) + PCondition 2 × (payoff for Action 2) + … + PCondition N × (payoff for Action N)
Of course, the payoff for each action may also involve probabilities. This sounds complicated, but it’s not difficult if you break it down.
- Figure out the payoff for each action, using probability if necessary, like we did above for the simple fighting strategy.
- Determine the probability of the condition that causes each action.
- Multiply the probability of each condition (from step 2) by the payoff of the action that it causes (from step 1).
- Add the probability × payoff pairs.
- To fill the payoff matrix, repeat this for each pair of strategies.
Let’s do this for the “fight to keep ownership of a resource, but don’t fight if someone else already owns it” strategy when paired against a simple “always fight” strategy.
Our conditional strategy has two possible actions, fight and not-fight. What is the payoff of each against a fighting strategy? We already solved the fight vs. fight payoff above, which is v/2−c/2. What about not-fight vs. fight? If we don’t fight, we simply get nothing and incur no cost of losing a fight, so that payoff is 0.