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This tutorial shows how to determine the average payoff to a strategy and how to state conditional payoffs that depend on probability. It assumes that you have a basic understanding of symmetric games from starting the Conflict I tutorial. If you work through all the examples in detail, this tutorial should take about 15 minutes.

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 P_{event}, so now we have:

P_{win} × (payoff for win) + P_{lose} × (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:

P_{Outcome 1} × (payoff for Outcome 1) + P_{Outcome 2} × (payoff for Outcome 2) + ... + P_{Outcome 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:

P_{win} × (payoff for win) + P_{lose} × (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:

P_{Condition 1} × (payoff for Action 1) + P_{Condition 2} × (payoff for Action 2) + ... + P_{Condition 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.

Now that we have the payoffs, we need the probability of each condition. The two conditions are owning the resource (in which case we fight) and not owning it (in which case we don’t fight). What is the probability of each? As with the fighting strategy above, we can make some simplifying assumptions. We’ll assume that every resource is already owned by someone. That is, it could be a territory that is already occupied, or it could just be that the first one to find a resource is regarded as its owner.

If every resource is owned, then there is an owner and a non-owner in every encounter, so the probability of an average individual being an owner is ½. Now it’s easy to calculate the total payoff:

- P
_{owner}× (payoff for fighting) + P_{non-owner}× (payoff for not fighting) - ½(v/2−c/2) + ½(0)
- v/4−c/4

So the payoff for the fight-if-owner strategy against the always-fight strategy is v/4−c/4.

70

To solve this, we first need the payoff for each possible action. Again, there are two actions, fighting and not fighting, and two conditions, owning and not owning.

If you own the resource, you will fight. Since your opponent also plays the fight-if-owner strategy, he will let you have the resource without a fight (because if you are the owner, he isn’t). So the payoff to an owner is simply v. There is no cost, since there is no fight. If you are *not* the owner, you won’t fight, and your payoff is simply 0. Again, there is no cost because there is no fight.

Now what about the probabilities? In each interaction, there is an owner and a non-owner. Since we want the payoff for an average player, the probability of owning or not owning is ½. So the total payoff is:

- P
_{owner}× (payoff for fighting) + P_{non-owner}× (payoff for not fighting) - ½(v) + ½(0)
- v/2

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How do we get this answer? If the fighter owns the resource, his opponent will not fight, so the payoff is v (with no cost). If the fighter does not own the resource, he fights the owner. If he wins, he gets v, but if he loses, he gets −c. There’s an equal probability of each outcome, so the payoff is v/2−c/2. (This is the same payoff that we found for fight vs. fight, and the reasoning is the same.)

Those are the payoffs, and once again the probability of getting each of them is ½, because the fighter and his opponent are, on average, equally likely to own the resource. The total payoff is:

- P
_{opponent is not the owner}× (payoff for not fighting) + P_{opponent is the owner}× (payoff for fighting) - ½(v) + ½(v/2−c/2)
- v/2 + v/4−c/4
- 3v/4−c/4

Now we have all the payoffs needed to fill in a matrix for the unconditional strategy always-fight and the conditional strategy fight-if-owner (remember that we did always-fight vs. always-fight at the start of this section):

always fight | fight if owner | |
---|---|---|

always fight | v/2−c/2 | 3v/4−c/4 |

fight if owner | v/4−c/4 | v/2 |

- The average payoff of a strategy is the sum of the payoffs for each possible outcome multiplied by the probability of each outcome. In the simplest case, this is P
_{win}× (payoff for win) + P_{lose}× (payoff for loss). - Payoffs are calculated on average for the whole population. That is, the payoff for a strategy is the average over all sizes and strengths of individuals. In general, you can reason about this by considering an average individual, who will be more able than his opponent half the time and less able half the time.
- If a strategy is conditional (e.g. act one way if larger and another way if smaller), first work out the payoff of each possible action and then multiply it by the probability that conditions will trigger that action.