Every row has a maximum payoff, and one of these maximum payoffs must be the overall maximum payoff , so a player always has a maximax strategy. However, several rows may be tied with the same overall maximum payoff, so a player may not have a unique maximax strategy. (The player’s several maximax strategies probably are not equally good with respect to other decision principles.)

(2) chose the row with the maximum of the minimums (the highest security level).

Every row has a minimum payoff, and one of these minimums payoffs must be the maximum of the minimum payoffs, so a player always has a maximin strategy. However, several rows may be tied with the same maximim payoff, so a player may not a unique maximin strategy.

(The player’s several maximin strategies probably are not equally good with respect to other decision principles.)

Maximize Average Payoff (“don’t focus on the best outcome or the worst outcome but on the average outcome”):

Every row has an average payoff, and one of these average payoffs must be the maximum average payoff, so a player always has a strategy that maximizes average payoff. However, several rows may be tied with the same maximum average, so a player may have several strategies that maximizes average payoff.

Maximize Expected Payoff (i.e., take account of the differing probabilities of contingencies):

(1) determine (or make a subjective estimate of) the probability p_k of each contingency c_k (where p_k > 0 and ∑p = 1),

(2) multiply each payoff by its probability and add these products together to get the expected payoff of each row, and

Note: average and expected payoff are the same in the event each contingency has equal probability.

For any probability distribution over contingencies, every row has an expected payoff, and one of these average payoffs must be the maximum expected payoff, so a player always has a strategy that maximizes expected payoff. However, several rows may be tied with the same maximum expectation, so a player may have several strategies that maximizes expected payoff.

Basic Principle: don’t choose a dominated strategy (or, always choose an undominated strategy).

It quite possible that no strategy dominates another, in which case every strategy is undominated. On the other hand, suppose s₁ is dominated Then there must be another strategy s₂ that dominates s₁. Remember that this means that s₂ has at least as great a payoff as s₁ in every contingency and a greater payoff in at least one contingency. Now suppose s₂ is dominated. Then there must be another strategy s₃ that dominates both s₂ and s₁, because domination (like “greater than”) is a transitive relation. If the number of strategies is finite, we must find (at least) one strategy that is undominated.

Two or more strategies may be undominated, in which case none is dominant. A player clearly can have at most one dominant strategy.

After the player has chosen his strategy and discovers what contingency nature has chosen, the player may regret his strategy choice and wish he had chosen a different strategy. Can you identify the condition under which a player would never have reason to regret his strategy choice?

For each contingency that may arise, the player has a strategy that is a best reply, i.e., that gives the best payoff in that contingency (column). (Two or more strategies may be “tied” in this respect, in that they all give same [best] payoff in that contingency.) A player has no reason to regret his strategy choice in a given contingency if and only if that strategy is a best reply in that contingency. A player has no reason to regret his strategy choice in any contingency if and only that strategy is a best reply in every contingency. You can verify that a strategy that is a best reply in every contingency is a dominant strategy. So the condition under which a player would never have reason to regret his strategy choice is when the player has a dominant and uses it.

In games against nature (as opposed to games against other rational players with their own interests):

(b) Maximin is unduly pessimistic (nature isn’t malevolent [even if it sometimes seems otherwise).

(c) Maximize average payoff probably doesn’t make sense because we always have some sense as to which contingencies are more and less probable.

(d) Maximize expected payoffs makes the best sense (provided that we have some sense as to what the probabilities are and the payoff numbers represent an appropriate kind of cardinal utility)

(e) It’s very hard to see anything wrong with principle that you should never use a dominated strategy (at least in a game against nature). The problem is that it’s a very weak principle that usually doesn’t tell you what strategy to choose. You can also check that the Dominance Principle (because it is so weak) never conflicts with any of the other Decision Principles, even the ones that don’t make much sense.

Preview to think about only (no written answer required). Suppose that “nature” is no longer a disinterested player but rather that nature is “malevolent” and is trying to hold the player’s payoff down to a minimum. Under this circumstance, do some decision principles become more justifiable? Less justifiable?

If nature is “malevolent” and is trying to hold your payoff down to a minimum, you are in a two-player zero-sum game. Expected Payoff Maximization is no longer directly usable, because your opponent will chose his strategy on the basis of rational principles, not probability. (However, as we will see, these rational principles imply that players should delegate their strategy selection to a chance mechanism in some types of [“non-strictly determined”] zero-sum games.) The Maximim Principle, which is too pessimistic in a game against nature (or non-zero-sum games with other players), is much more justifiable in such a context of total conflict. And the Dominance Principle (for what’s worth) still makes sense in a zero-sum situation.

The concept of the best reply is relevant to the question in the following problems that asks whether it is possible to find a probability distribution over contingencies that makes a specified strategy an expected payoff maximizing strategy. If a strategy is a unique best reply (gives a payoff in that contingency that strictly higher than the payoff given by any other strategy in the same contingency), there is an “extreme solution” that answers the question positively. That is, a strategy is expected payoffs maximizing under any probability distribution that make the contingency in which the strategy is the uniquely best reply almost certain to occur makes

This raises the question of whether a strategy that is not a best reply in any contingency can maximize expected payoffs. The answer is that it can, if the strategy, although not best in any contingency, is “good” in many or all contingencies. Here’s a simple example:

	c₁	c₂	c₃
s₁	7	3	4
s₂	5	5	5
s₃	2	5	7

Note that s₂ is not a (unique) best reply in any contingency but is a “good” reply in all. Indeed, s₂ has the highest average payoff and thus is expected payoff maximizing for the probability distribution ⅓, ⅓, ⅓ (among others). (Strategy s₁ in II is of this character.)

However, if a strategy fails to be a unique best reply in any contingency because a single other strategy is at least as good in all contingencies and better in at least one (i.e., if the latter strategy dominates the former), the former dominated strategy has a lower expected payoff than the latter strategy under any probability distribution. A corollary is that a dominant strategy maximizes expected payoffs under every probability distribution.