In managing risky activities, decision makers (DMs) may have different available choices, payoffs or utilities (see Utility Function) for outcomes, as well as cultural expectations and ethical or moral values constraining how they believe they should behave. For example, a confrontation occurs in an international forum, such as in the development of certain terms for a convention or a treaty, where some stakeholders want to determine their best payoff, while minimizing that of another, before or during their negotiations (see Considerations in Planning for Successful Risk Communication; Role of Risk Communication in a Comprehensive Risk Management Approach). Similarly, an industry may assess what it can pay, before a fee or penalty schedule is codified. In these and many other situations, it is advantageous to describe alternatives or choices; positive or negative outcomes associated with each choice; probable effects of each choice, modeled by evaluating the probable consequences of different combinations of choices made by the DMs or stakeholders. The study of strategies and counterstrategies clarifies what a solution to a conflict (or partial conflict) situation is, when one exists, whether it is unique, why and whether it is stable (so that it is not plausible for players to deviate from it), conditions under which communicating with an opponent is or is not advisable, and ways to systematically change assumptions to test the sensitivity of results to those changes.

This article deals with key aspects of decisions by one or two stakeholders that confront alternative choices. In this confrontation, a strategy is a feasible act or option (including decision rules specifying what to do contingent upon information available at the time a choice must be made). Each stakeholder is assumed to have a discrete set of possible strategies that are mutually exclusive and fully exhaustive; these strategy sets are assumed to be common knowledge. A measure of value associated with each outcome is the payoff that each of the individuals has. Payoffs can be measured in units of money (usually transferable), utility (not necessarily transferable among players), or physical units (e.g., number of deaths or life-years lost), depending on the application.

1 Framework for Analysis

1.2 Two or More Decision Makers with Different Objectives

To depict a situation in which parties confront each other, one can replace “Nature” in the preceding example by a second rational, deliberating player. Options and payoffs for each player can still be tabulated. For example, here is a 3 × 2, read three-by-two, representation (a matrix) of a hypothetical confrontation between two players, imaginatively called Row and Column (blue and red are also often used, especially in military applications).

The table shows Row's payoffs, measured in some units, such as dollars or utilities. If Row chooses the first row and Column chooses the first column, then Row receives a payoff of four units. In this type of zero-sum game, whatever Row wins, Column loses (and vice versa); thus, Row prefers large numbers and Column prefers small ones.

The largest of the row minima (maximin) is 4, and the smallest of the column maxima (minimax) is also 4: thus, the pair of choices (r₂, c₂) has the property that neither player can do better by deviating from (i.e., unilaterally choosing a different act from) this pair. The fact that the minimax (also called the security level for a player) and the maximin are the same identifies a “saddle point” solution (since, like the lowest point on the long axis of a saddle-shaped surface this point is simultaneously a minimum in one direction—it is the smallest payoff in row r₂, and hence the smallest one that Column can get when Row selects r₂—but is a maximum in the other (largest number in its column, and hence is the largest number that Row can achieve when Column selects c₂)). This is the “best” solution for both players, in the sense that neither can obtain a preferred payoff by a unilateral change of strategy (i.e., Row cannot do better by choosing a different row; and Column cannot do better by choosing a different column.) The celebrated minimax theorem (1) shows that any zero-sum game has a saddle point solution, provided that players are able to choose among alternative acts using any desired set of probabilities. In many practical situations, however, there is no saddle point among “pure” (nonrandomized) strategies. For example, consider the following 2 × 2 zero-sum game:

Because the minimax differs from the maximin, this game does not have a saddle point among pure strategies. Further, no strategy dominates the other for either player (i.e., no strategy yields a preferred payoff for all choices that the other player could make). Nonetheless, if payoffs are measured using NM utilities, then a minimax strategy necessarily exists among mixed strategies.

Numerical solution techniques for zero-sum games (e.g., linear programing for general zero-sum games (Luce and Raiffa [(2), Appendix 5]), Brown's method of fictitious play (Luce and Raiffa [(2), Appendix A6.9), the method of oddments (Straffin (3) and Williams (4)), and various geometric methods) are well developed. For the game,

a simple geometric technique suffices: the two payoff lines represent Row's optimal probabilities for mixed strategy. The approach is depicted in Figure 1, as seen by Row, where the vertical axes measure Columns' payoffs and the oblique lines show the strategies for Row.

The solution (obtained algebraically) is 0.375 and thus Row 1 should be played with probability 0.375 and Row 2 with probability 0.625. Column will play strategy 1 with a probability that equals 0.75 and strategy 2 with probability that equals 0.25. Observe that the original game did not have a saddle; the approach requires assessing that the 2 × 2 game does not have either a saddle or dominance by a strategy, and then proceed as developed. The importance of this technique is that it allows for the solution of n × k zero-sum games by assessing mixed strategies in 2 × 2 subgames of the original n × k game (a result that is guaranteed by the minimax theorem (Straffin (3)), due to John von Neumann).

1.3 Beyond the 2 × 2 Game

A DM can use alternative criteria for choosing one course of action over another. We further discuss the minimax, maximin, and expected value using a situation in which Row now faces Nature in a 4 × 4 game (Straffin (3)):

There is no row dominance. That is, no row has values that are at least as great in every column (and strictly greater in at least one column) as the values in all other rows. However, column 2 dominates all other columns. The minima of the rows are [0, 1, 0, 0] with Row's maximin equal 1. The maxima of the columns are [2, 4, 1, 1]' with Column's minimax equal 1 utility (' denotes transpose, i.e., writing a column as a row or vice versa). There are two saddle points at r₂, n₃ and at r₂, n₄. If Row believes that all four columns are equally probable, then the strategy with the largest expected utility is r₁, with E(r₁) = 0.25 × 2 + 0.25 × 2 + 0.25 × 0 + 0.25 × 1 = 1.25.

We can also calculate the regrets for Row, defined as the difference between the best payoff and the alternative payoffs, conditioned on each n_j, as shown below:

In this case, the row maxima are [2, 3, 2, 1]' and, if we were to adopt the minimax regret criterion, then the optimal choice for Row by this criterion would be r₄. If the expected regret was calculated using equal probabilities (pr) for the four states of nature (pr(n_j) = 0.25), then the result would be ambiguous (they would be r₂, r₃, and r₄) because there are three equal and largest expected utility values.

Because different decision criteria lead to different optimal choices, several axioms have been developed to guide their selection. Some of those axioms are summarized in Table 2 and are related to the criteria for justifying that choice.

Table 2. Selected Axioms for Decisional Criteriaa

Axiom	Description	Criterion met by	Criterion not met by
Symmetry	Permuting rows or columns should not change the best choice	Maximin, expected value, regret	NRb
Dominance	When every row element is greater than another, the latter is inferior and can be ignored	Maximin, expected value, regret	NR
Linearity	Multiplying by a positive constant and/or adding a constant to all entries in the payoff matrix should not change the best choice	Maximin, expected value, regret	NR
Duplication of columns	Repeating columns should not change the best choice	Maximin, regret	Expected value
Addition of row	Repeating row or rows should not change the best choice	Maximin, expected value	Regret
Invariance to a change in utility	A constant amount of utility or disutility does not change the choice	Expected value, regret	Maximin
Comparability	Adding a weakly dominated row should not change the best choice of act	Expected value, regret	Maximin

• ^a Adapted from Straffin ⁽³⁾
• ^b Not reported

2 Searching for a Common Ground: Nash Equilibrium

The process of understanding rational decisions can be shown by the same normal form representation of strategies, S_{i, j}, consequences, C_{i, j}, and probabilities adopted in the previous section with the added understanding of the possibility of choices of strategies that are optimal. We use 2 × 2 games in which DMs can either cooperate or not. Some games do not allow sequential choices, while others do. Strategies, as course of actions, say cooperate, defect, attack, retreat, and so on, can be either pure (no probabilistic assessment is used: deterministic numbers are used) or mixed (probabilities play a role in assessing the eventual outcome from the situation confronting the DMs). A 2 × 2 game has two strategies per DM: for example, either defect or cooperate, but not both. We will discuss games that are effectuated simultaneously; further information is given in Burkett (5).

2.2 Mixed Strategy Games

The distinguishing feature of mixed strategies is that each is characterized by probabilities. Continuing with the 2 × 2 representation, if strategy for DM1 has two outcomes, then one outcome has probability pr and the other has probability (1 − pr). For the other DM, however, we label the probabilities as q and (1 − q). Consider the earlier game as in Table 5.

Table 5. Example for Mixed Strategy (also see Table 3) Using Arbitrary Probabilities and Leading to a Unique Nash Equilibrium

		Decision maker 1
		S1	S2
Decision maker 2	S1	4, 4	0, 6
	S2	6, 0	3, 3

We can calculate the expected payoffs as follows:

• For DM2 and S₁: (pr)(4) + (1 − pr)(0) = (pr)(4). For DM2 and S₂: (pr)(6) + (1 − pr)(3) = (pr)(3) + 3.

• For DM1 and S₁: (q)(4) + (1 − q)(0) = (pr)(4). For DM2 and S₂: (q)(6) + (1 − q)(3) = (q)(3) + 3.

Putting arbitrary probabilities (0.5) in these expressions, we obtain:

• For DM2 and S₁: (0.5)(4) + (1 − 0.5)(0) = (pr)(4). For DM2 and S₂: (0.5)(6) + (1 − 0.5)(3) = (0.5)(3) + 3 = 4.5.

• For DM1 and S₁: (0.5)(4) + (1 − 0.5)(0) = (0.5)(4). For DM2 and S₂: (0.5)(6) + (1 − 0.5)(3) = (0.5)(3) + 3 = 4.5.

It follows that the game characterized by the unique Nash equilibrium remains unchanged. This is not the case for the other example that did not have a Nash equilibrium (Table 6).

Table 6. Example for Mixed Strategies Showing Indifference to Each Other's Choice

		Decision maker 1
		S1	S2
Decision maker 2	S1	Profits = 1, does not profit = 0	Does not profit = 0, profits = 1
	S2	Does not profit = 0, profits = 1	Profits = 1, does not profit = 0

The mixed strategies approach changes the deterministic results. Putting arbitrary probabilities (0.5) in these expressions, we obtain

• For DM2 and S₁: (0.5)(1) + (1 − 0.5)(0) = 0.5. For DM2 and S₂: (0.5)(1) + (1 − 0.5)(0) = 0.5.

• For DM1 and S₁: (0.5)(0) + (1 − 0.5)(1) = 0.5. For DM2 and S₂: (0.5)(0) + (1 − 0.5)(1) = (0.5)(3) + 3 = 0.5.

The result is Nash equilibrium for these probabilities: both DMs are indifferent to the strategy of the other.

The concepts of sequential games (5), where the choices made by the DMs account for the evolution of the choices, can be illustrated (Figure 2) as a decision tree in which the lines are the branches of the tree:

Each branch identifies a possible strategy that can be taken by a DM. This tree identifies simple binary choices and payoffs, which can be positive (+), negative (−), or zero (0). Reading the tree from left to right leads to an understanding of how the two DMs can behave and their possible final choices. DM1 would probably do, while DM2 might try to change the game, if it were possible.

3 Discussion

In the end, it is preferable to have more information than less (not necessarily, e.g., mutual insurance can be destroyed by free information, reducing everyone's expected utility); thus, anticipating the outcomes of a dispute provides useful information by orienting and informing a choice.

How does this theoretical framework compare with actual behavior? The basis for game theory is that DMs optimize their choices and seek more utility, rather than less. Although this basis is theoretically sound, there have been many cases of violations in simple, practical situations. Some, for example, Kahneman and Tversky (1979) found that actual behavior affects decision making to the point that the theory, while sound, required modifications, thus resulting in prospect theory. Camerer and Fehr (6) have also found that DMs inherently act altruistically and thus are not money maximizers.

Glimcker (7) reports the following. He uses the game work or shirk in a controlled setting: the employer faces the daily decision of going to work; the employer has the choice to inspect the employee's performance. For example consider Glimcker [(7)] (with change in terminology). The two stakeholders incur daily monetary costs and benefits in unstated monetary units, as shown in Table 7.

In general, because inspection cost is high, the employer will be allowed some shirking. If the inspection cost was lower, they could result in more inspection and thus lead to decreasing rates of shirking: the result is Nash's equilibrium. For the employer, the hazard rates are (shirk) = ($inspection/$wage); (inspect) = ($leisure/$wage). The Nash equilibrium is (the result is obtained by solving for the probabilities, e.g., 50 × x + (1 − x) × 50 = 100 − 100 × x; thus x = 0.5) shirk 50% of the time and inspect 50% of the time [(7), p. 304]. Students (unfamiliar with game theory) were enrolled in the shirk–inspect game to assess if their behavior was consistent with this theoretical equilibrium. In the first 100 (out of approximately 140) tests, the stakeholders modified their outcomes and tended toward the Nash equilibrium. When the payoffs were changed, the results changed as well.

The shirk-work game discussed in Glimcker (7) provides some interesting insights on the sequential choices made between either working or shirking, when deciding which alternative to take by the worker who is initially contemplating the choice of either working or shirking work. The results show that the worker (in an experiment involving 150 trials) appears to be acting randomly and thus keeps the employer guessing as to what course of action she will actually take. This author reports that the choices, made in 150 games, has the results as in Figure 3.

The decision tree shows the chronology of the decision. The initial choices are to either work or shirk. The second decision, conditioned (the conditionalization is symbolized by the symbol “|”), is to work | work, shirk | work, and so on, as depicted by the tree. It appears that the worker is acting randomly and thus keeps the employer guessing!

Clearly, individual choices cannot be divorced from the fact that those choices are made in the brain. Interestingly, it is now increasingly apparent that humans have (8) … at least two systems working when making moral judgments …. There is an emotional system that depends on (the ventromedial prefrontal cortex) … and another that performs more utilitarian cost–benefit analyses which in these people is intact. When that region is damaged, individuals opt for choices that are different from those not damaged. For example, in a relatively small sample of respondents, only approximately 20% of normal individuals answered yes, while approximately 80% of individuals with damaged brain answered yes, to the following question:

You have abandoned a sinking cruise ship and are in a crowded lifeboat that is dangerously low in the water. If nothing is done it will sink before the rescue boats arrive and everyone will die. However, there is an injured person who will not survive in any case. If you throw the person overboard, the boat will stay afloat and the remaining passengers will be saved. Would you throw this person overboard in order to save the lives of the remaining passengers?

Now consider a different choice (Carey (8)): having to divert a train by flipping a switch, to save five workers, knowing that such diversion would kill one other worker for sure. The research shows that those with the brain injury, normal individual, and those with a different type of brain injury would divert the train. When that certainty of killing the single worker was not apparent, all of the three groups rejected the trade-off. Moreover, if the action did not comport flipping a switch but, rather, the act was equivalent to pushing that single individual to certain death to save several others at risk, the results by all of the three groups were still different: those with the ventromedial injury were about twice as likely as other participants to say that they would push someone in front of the train (if that were the only option …). Although the brain injury is uncommon, the responses may make some think twice about the implications of either answer.