Game theorists have not paid much attention to emotions. This seems odd, since our theory's object of study is social conflict, where emotions are sure to arise. Nor has anyone given a good account to justify the neglect, although some writers have hinted at reasons. In a recent debate on rational choice in the New York Times (Feb. 26, 2000), Morris Fiorina suggested that the parties become more rational when the decision becomes more important. The implication is that our methods should at least succeed for the more important issues. However, one could argue that the more serious the decision, the more it will generate emotion. In my own area of international conflict, it would be hard to show that the big decisions for war or peace were made calmly.

Game theory needs an account of emotions, and it needs one that goes beyond the folk concept. It should base its approach on the findings of psychology, anthropology and philosophy, which offer an extensive literature on the subject. The goal of this paper is to survey some treatments that game theorists have already proposed, and compare them with a more full concept of emotions.

The promising approaches fall into four categories:
- models of surprise in emotion using psychological games
- emotions as equilibria using modified utilities, based on the goals or moves of others
- depiction of an emotional player as a version of the original with a different utility function
- depiction of an emotional player as selecting a move at random with probabilities determined by the utilities.

The conclusion will be that some existing approaches are on track but they must be developed. Another is that different emotions will require quite different methods.

I will look only at how game theory might portray people as acting emotionally. This is different from those who emphasize the survival value of emotions (e.g., Hershleifer, 1988, 1993; Frank, 1988; Morrison 1996), and this subject will not be treated. It is also different from studies of what emotions might arise in the course of play (e.g., Brams, 1997, on frustration and anger.) Nor will I look at the role of emotional displays as credible signals in games -- the idea that anger is a visible proof of resolve, or a blush shows we are truly sorry. Also I should note that some papers I discuss have not set their main goal as treating emotions, so if they have failings in this regard, these may simply reflect the authors' intentions.

What is an emotion?

Articles and books continue to appear with titles like "what is an emotion?" more because of continuing disagreement than accumulating knowledge. This section will describe some features of emotions that strike me as reasonable, recognizing that just about every one of them would be disputed by some expert.

First, an emotion is a kind of mental sensation. The phrase "mental sensation" is meant to be non-redundant - the sensation must be located in the mind, rather than in some part of the body or in the external world. Hunger or the perception of the colour blue, for example, are not emotions.

Since the emotion is the sensation, the individual is the final authority on it. To feel happy implies that one believes one is happy, and vice versa. This criterion leads to a useful way to exclude certain words as not referring to emotions - the "be/feel" test (Clore, Ortony and Foss, 1987). When these two verbs can be substituted without changing the meaning, the concept passes a necessary test for an emotion. To feel happy and to be happy are the same. To feel rejected and to be rejected are different. To feel proud is an emotion, as is to feel alienated from one's work, but to be ignored is not. The test seems easy to apply and usually gives acceptable results. Still, there are borderline cases: surprise, curiosity, amusement, frustration or lust.

Emotions are passive - they happen to us. They relatively short term. Just how short they are may depend on the strength of the stimulus, but if something lasts for days it is better called a mood.

A further way to understand the concept is to look for a structure within the set of emotions. One approach is to identify some of them as "basic." For example, Oatley and Laird-Johnson (1988) specify five: happiness, sadness, fear, anger and disgust. These are regarded as primitive, i.e., they cannot be further defined and they need not be. The non-basic emotions can be defined in terms of these five, sometimes by adding stipulations about strength, or about combinations, or about the cause. Loathing is a blend of anger and disgust. Confidence is mild happiness caused by a belief that one will succeed at a coming task. Gladness is happiness based on the belief that a certain desired event has occurred.

Specifying the perceived cause is a common way to define an emotion, and is one sense in which emotions have a cognitive component. Usually the cognition involves the difference between the perceived world and some comparison world. We are angry not just because someone actually injured us, but because we have in mind the proper and fair world where they did not. We experience gladness because what happened was better than what we expected. Interesting research has been done on how we generate the comparison world (Kahneman and Miller, 1986) - it is often done after the outcome is known, based on learning the outcome itself. We head for the airport, get stuck in traffic and are delayed by half an hour. We arrive there to learn that we missed our plane, which left on time, 30 minutes ago. Compare our reaction with a situation where we learn that we missed the plane, but it left 25 minutes late, just five minutes ago. The second case is more distressing and disappointing because of a fact we learned only after we knew the outcome. This aspect of emotions, that they compare the actual world with another hypothetical one, is amenable to treatment in game theory. Some path of play will be the reference in the player's mind. The choice of this path may involve the player's expectation or norms or some other criterion.

As well as a system of basic and defined motions, categorizations have been proposed. The goal is to include many emotions in a few groupings with clear boundaries (e.g., Ortony, Clore and Collins, 1988; Ben Ze'ev, 2000). A basic division is positive versus negative - whether we like the current situation. A further distinction separates an emotion about the action of an agent, the fortune of an agent or some event without an agent. Of course, the agent can be oneself. Pride is a positive response to one's own action.

The next step is a theory of how emotions function. One idea regards them as signals from our more subliminal and intuitive side to our cognitive mind. The premise is that conscious processing of decision variables is a scarce commodity. Our logical mind easily gets overloaded and confused. Also, it is too slow to do all the work -- we cannot search down each part of the decision tree. Emotions are fast, functioning on the level of associations and similarities. An emotion signals that a certain branch is good or bad, and induces us to explore it further or snip it.

Damasio (1994, quoted by Elster, 1998) strikingly illustrates this idea with a patient who suffered brain damage depriving him of emotional feelings. Life became flat, and while he recognized that he had suffered a tragedy, he talked about it freely without any sign that he minded. On one occasion his therapist asked him whether Wednesday or Thursday would be better for his next appointment. He examined his book and began to list the costs and benefits of each day. After a full half hour, the therapist interrupted him, "How about Thursday?" "That would be fine," he said, and left. Emotions avoid this kind of excess, by telling us that certain paths do not feel right, so we should look elsewhere. They are necessary supplements to reason.

When emotions are mild they prune our decision trees, but when they are stronger they take control of our attention. When we are angry or happy or in love or in grief, our minds are drawn to the object of the emotion. Certain of its features become more salient -- the person's good aspects (for love) or bad ones (for anger) or the circumstances of the loss (for grief). As well as affecting our judgement and attention, a strong emotion affects our utilities. When we are abused by some store clerk we are ready to set our afternoon's work aside to write a letter of complaint. Our intention lasts as long as our anger.

It can be argued that this arrangement has evolutionary value. Sometimes it is crucial to flee, to fight, or to stop and focus on what is happening, and we are forcibly made afraid, angry or sad. From a survival viewpoint, certain emergency situations are too important to trust to cognition.

With this characterization of emotion we can turn to some game theory approaches.

The lexicographic approach

Nalebuff and Shubik (1988) discuss the emotion of revenge, (although I would term it "anger" to satisfy the "be/feel" test.) At Princeton Shubik and his colleagues had played a vicious card game "So Long Sucker"(Hausner, et al., 1964), with the following general feature: the contest of several people would eventually leave only one winner. The parties must sequentially cut individuals out, and an eliminated player is able to give his resources to some of the remaining players. He cannot win himself but his desire for revenge will work against someone else, so no one will want to do the dirty work of eliminating him.

Their approach is "lexicographic" in the sense that revenge becomes important only when the player cannot win, so the courses of action are equal for the player's self-interest. An emotion is here a kind of self-indulgence, recalling Fiorina's idea that players act on them mostly when it does not matter. However, I do not see this as accurate about human nature -- emotions are more prevalent and more consequential.

Emotions and surprise -- psychological games

Psychological games, introduced by Geanakoplos, Pearce and Stacchetti (1989), have often been used to portray emotions. Their innovation is to allow the equilibria to depend not just on the outcome, but on players' beliefs about the play of the game. This means more than adding a term into the payoff function, since those beliefs normally arise endogenously as a result of equilibrium considerations. The definition has some unexpected consequences - the equilibrium of trembling hand perfection, for example, may fail to exist.

The authors illustrate their notion using gift-giving, and this idea is later developed by Ruffle (1999). His goal is to understand why people give gifts, rather than simply transfer money, which the recipient can spend in the most efficient way. His explanation is the emotions induced by the gift -- surprise and appreciation for the receiver, and pride for the giver.

A numerical example of his simplest game is given here. The gift costs the giver 1 unit, and benefits the receiver 1 unit. Also there is an emotional factor. Suppose the receiver expects the gift with probability p, otherwise nothing. The measure of pleasant surprise and appreciation at getting the gift is given by an additive term 1 - p in the utility. Similarly if no gift comes the unpleasant surprise term is -p. The giver is motivated by pride and receives an increment of

2(1 - p) for giving the gift but loses -p/2 for the embarrassment of not doing, which is higher the greater the receiver's expectations. These values are based on the assumption that the giver has a correct view of the receiver's expectations, p.

All told, the giver's and receiver's respective utilities are:

Give the gift:   -1 + 2(1 - p), 1 + (1 - p)

Not give it:       0 - p/2,          0 - p

The equilibrium in this simple version will depend only on the giver's values. There is no pure strategy equilibrium, but p = 2/3 is a mixed strategy equilibrium.

The persuasiveness of this idea depends on how one interprets mixed strategies. If they are probability settings on a random device - if the giver is supposed to flip a biassed coin - it is not clear why the giver should feel proud that the coin turned up heads rather than tails. Also, it is not clear why the receiver, recognizing this as the basis for the gift, should appreciate receiving it. If we interpret p in a more modern way, as the receiver's subjective belief that the giver will give the gift, there may be a plausible argument -- the first player was actually going to give the gift all along, but the second did not know it, so the former reveals his intention and has a right to feel proud. However, the conceptual argument involving mixed strategies needs to be made explicit.

Psychological games tap an important aspect of many emotions, the ones that depend on learning information, such as gladness or disappointment or relief, but in my view the models would be improved by using games of incomplete information. In social situations we feel anger or appreciation not when we learn the outcome of a random variable, but when we learn something about the other player, that their loyalty or thoughtfulness is lower or higher than we thought. This calls for incomplete information, but so far psychological games have been defined only for the complete information case.

Emotions as equilibria with modified utilities

Rabin (1992) has developed a theory of fairness which he describes as involving motives of emotion. He applies the notion of psychological games, but a player's interest is not in being surprised. Each is concerned about the belief of the other player that lies behind the choice of a move.

Consider the example of O. Henry's story "Gift of the Magi." Several authors - Rapoport, Vorobev and Rasmussen - have construed it as something like the following 2x2 game (although Rabin himself does not claim the game as an example of his theory.)
 

We will interpret the 2's in this matrix as the objective payoff each partner gets when the other has a "full" set of possessions. They do not represent altruism. The equilibria in the game give the outcomes (3,2) and (2,3).

In Rabin's notion each is deriving benefit from believing that it is responding appropriately to the other's move, in the sense of rewarding kindness with kindness, and vice versa. Whether the other is being kind to one depends on its intention in using the particular strategy. Suppose the man

(0) decides to play Keep,

(1) believes the wife has decided to play Keep, and

(2) believes the wife believes he has decided to play Keep.

The last two beliefs suggest that he thinks she is trying to hurt him, and in this case he attaches benefit from hurting her. Indeed he is hurting her by playing Keep. Her logic is the same. If each plays as in (0), and believes as in (1) and (2), their beliefs are correct and their actions are maximizing given their beliefs. They are maximizing toward the goal of material interests combined with fairness in the sense of reciprocity. (Rabin describes a formal way of combining these two.) The model portrays the emotion of anger or indignation.

A related approach is by Segal and Sobel (1999) whose players hold the usual preference ranking over the outcomes, and another one over their own strategies. The latter is specific to the game being played and to the strategy that the other player is using. If you are nice I prefer one of my strategies to another, but otherwise I might have the other preference. The authors introduce certain axioms that indicate the model is equivalent to holding a utility over strategies that is a linear weighted function of the payoffs, where the weights depend on the strategy the other is using. In O. Henry's game, for example, suppose that if the woman is playing Sell, the man's utilities for the payoffs are v_m + 2 v_w, where v_m and v_w are his and her material payoffs in the original matrix. If the woman is playing Keep it is v_m - 2 v_w. The woman's payoffs are analogous. This transforms the game to

An equilibrium is again mutual Keep.

O. Henry advises us in the end that when they ended up in the upper left cell, they actually both won. We would not expect Rabin's or Sobel and Segal's approach to produce O. Henry's story, since both parties, and the reader, were surprised by the outcome.

Levine's approach (1997) is philosophically close to these. It has A's utility for the material benefit of B depend on A's estimate of the weight B is giving to A's benefit. (Neither Levine or Segal and Sobel describe their work as addressing emotions.)

These extensions are promising in some ways, but are somewhat off target as models of emotions. It seems right that the players become emotional in response to the moves of others. The models complement the previous "surprise" approach in that they deal with those emotions that are not surprises. Both players have full foresight about the other's intentions before the move is made. They have a thorough understanding of why they are in equilibrium at the outcome. However this is a problem for these theories as analyses of emotions. The husband realizes why the wife is moving as she does -- she is punishing him just because he is punishing her. Often anger is supported by wearing mental blinders, not thinking deeply or symmetrically about the situation. At least we would expect some claim about who became hostile first. However they are indignant at the other simply for doing what they themselves are doing, and they realize it. The logic of equilibrium analysis is at odds with nurturing the emotion.

Emotions and self-manipulation

A contrasting approach would say that if the couple have the foresight to choose an equilibrium at all, they should have the foresight to avoid that particular one. It would recognize that our emotional selves have different goals than our normal selves, goals which our normal selves may not want to promote. I may feel satisfied in telling someone off when I am mad at them, but I do not seek out situations to make me angry in order to get that satisfaction.

The next model expresses this idea. A player comes in several versions and control goes from one to another during the game. The earlier version of the player sets up the situation in a favorable way to himself before exiting.

Consider the following problem. I have a chocolate bar, which I can eat in parts over three days. I want to choose x₁, x₂ and x₃, with x₁ + x₂ + x₃ = 1, to maximize u(x₁) +   u(x₂) + ² u(x₃). With a discount factor_i = .7 and a daily utility function which is the square root of x_i, for example, I can calculate the following plan as best: I eat .578 today, .283 tomorrow and .139 on the third day.

This division maximizes my total discounted utility, calculated today. However, note that when tomorrow comes, I will have .422 left, and I have to ask myself: Will I be willing to divide that amount up tomorrow in just the proportions that I plan on today? If not, then my current calculation is an illusion. In fact, however, I can trust myself to follow the plan just because of the form of the discount function. Constant discounting produces consistency over time (Strotz, 1956).

Let us modify the example by introducing a mood of depression. The less chocolate I have on a day, the more depressed I become, and this is reflected in my discount factor. When I am depressed I am too discouraged to provide for the future and the discount factor goes down. We can model this by letting _i on day i equal simply x_i, the amount of chocolate I eat that day. Both will lie in the interval 0 to 1. In this variable-discounting case, I may make a three-day plan that is optimal for me now, but I cannot trust myself to carry it out, so I must strategize about myself - I must leave an amount of candy that my tomorrow-self will divide up in a way that does as well as possible for today's longterm goal. Calculations show that I can maximize my total discounted utility from today's viewpoint by eating .769 today. My tomorrow self will respond to the remainder by eating .222 and leaving .009 for the third day. The proportion saved drops sharply, since I get more depressed and give less weight to the future.

The possible divisions for various discount rates, and the division with chocolate-deprivation depression, are shown in Figure 1. Note that the depression division lies off of the curve of the fixed-rate divisions. This means that no matter what discount rate I have on the first day (i.e., how much candy I eat), my future self will not divide the remainder in the proportion optimal for that discount rate. It is truly a three-player game.

This model takes after game approaches to dynamic consistency in decision making (Pollak, 1968). It is an odd model of emotion overall - I do not mind being depressed except in that it influences decision making about consumption -- but it reproduces one frequent feature, that a person given to emotions must engage in self-manipulation. This is common in life. I may stay away from some annoying person for fear of what I might do if I am angry. I may avoid reading my end-of-class evaluations for fear of becoming discouraged, or I may not want to hear some bad story about someone I like. This goes against the behavior of a good utility-maximizer, who is never averse to more information (Wakker, 1989).

Emotion as loss of control

A frequent metaphor for some emotions involves an explosion, often of a hot liquid in a container. Someone becomes agitated, gets hot under the collar, then blows up, loses it, boils over, or flips his lid. A related metaphor is insanity - he goes nuts, bananas, apeshit (Kovecses, 1990, 2000.) This idea is captured in two models, the first where the loss of control is complete, and a better version where it is partial.

"Explosive" bargaining: Two negotiators must divide a dollar. If they fail to agree, they get none of the dollar, and also receive penalties of anywhere from $0 to $1. Each one knows its 

own prospective penalty and holds a uniform distribution over the other's penalty. The feasible set and possible disagreement points are shown in Figure 2.

The game has a twist. Before making a demand, each can say something that may make the other mad and prompt them to walk out. It might be stating a high demand, or delivering an insult. The action is chosen from a continuum, and indexed by the probability the other will walk out. (In the game the strength of the insult will function as a credible signal of one's own type.)

The rules are as follows:

Stage 0: Each learns its disagreement payoff, t_i.

Stage 1: Each insults the other, in effect choosing the probability in [0, 1] that the other will walk out.

Stage 2: If both are still present, each announces a demand x_i, in [0, 1]. If the demands are compatible (if x₁ + x₂ 1), then each receives its respective demand. If not, each gets its disagreement payoff.

I will look at equilibria of a certain form, where the players reveal their disagreement payoffs by their strategies at stage 1, and where their stage 2 demands equalize their increments over these disagreement payoffs, which they then know. There is a unique separating equilibrium of this kind. Each uses an insult of strength .3936 (t_i + 1). Thus those with the greatest penalty do not insult the other at all, while those who are in the best position, signal their strength by generating about a.40% risk that the other will leave. They do not deliver the worst insult, since even they have something to lose.

Partial loss of control. The explosive model incorporates the loss-of-control feature of emotions, but it is too extreme. People's likelihoods of walking out of the session should depend on what they might gain by staying. Real humans do not lose control completely. Anger does not turn us into raging maniacs, in general. Our perceptions, judgement and goals may be distorted, but we continue to keep an eye on our own gains and losses, and we also consider the other's losses. In the prototypical notion of anger, we "get even" but do not commit unlimited destruction.

Selten incorporated partial control ideas into a model of kidnaping. A kidnapper announces a ransom demand, and the family then decides how much to pay, which may be less than the full demand. The kidnapper then decides whether to let the hostage go and take the money, or commit murder. Emotion enters in at this final decision of the kidnapper. He is frustrated by not getting his full demand, and yet wants to take at least the partial payment and avoid a greater penalty if caught for murder. The kidnapper's choice of release or murder is a random one, but its probabilities are determined by the relative payoffs of each course.

Note that the model has the element of self-manipulation like the chocolate-eating example. In choosing the demand, the kidnapper looks forward to his own future random behavior when the family offers him partial payment. He chooses it to maximize his current utility, which may be lowered by his action under anger.

In O'Neill (1989) I applied essentially this model to formalize NATO's policies about deterring a Soviet attack on Europe. NATO chooses its arms levels, considering the fact that it will suffer more or less should it be induced to retaliate, which would be, like the kidnapper's move, a random act. The size of the arsenal will cause more or less harm to the Soviet Union and influence its probability of retaliation. The game involves incomplete information about Soviet goals. Both NATO's retaliation and the Soviet response involved random moves, but the analysis was still tractable.

The actor becomes a random device, with probabilities influenced by utilities. The essence of the argument is shown in Figure 3. In the usual game, on the left, Player 2 could retaliate and lower 1's payoff from a₁ down to r₁, but that would hurt 2 since its retaliation payoff is also lower than its acquiesce payoff. The normal outcome would be Grab and Acquiesce. However, we can introduce the probability of retaliation. Letting n be the non-compliance damage s₂ - a₂, and r be the retaliatory cost a₂ - r₂, a reasonable family of functions for the probability of retaliation is

p(n, r) = [n/(n + r)]^e/(1-e), where 0 < e < 1. The parameter e measures the degree of self-control. For e near 0, the retaliator is essentially ignoring the consequences of his actions, and in the limit of e = 1 it corresponds to the regular model.

This probability-of-retaliation function is not arbitrary. Considerable work has been done on justifications for random utilities (Fishburn, 1998), and here the various moves are chosen according to probabilities that satisfy one particular model, Luce's constant ration rule.

This concept of the emotion of anger suggests that one forsees how angry one will become (the parameter e is known), but does not know what one will do. This makessense given that an emotions forcibly enter into our judgements, perceptions and values. When we are calm we are not our emotional selves, and do not know how we would then perceive the situation and act.

Conclusion

The lack of a theory of emotion has produced some odd distortions in game theory. It has often been credited (or blamed) with devising nuclear strategy, and although this is an exaggeration there was a great literature produced on the subject during the 1980s (O'Neill, 1995). However emotion is an essential element in this context of deterrence (Crawford, 2000), since it is hard to devise a real reason for retaliating after an attack, especially if the other has kept some forces in reserve. Policymakers' may not say so, but the rationale for their strategies depends on a role for anger. Game theory analyses generally stayed silent on this element, or altered the real problem so it no longer played a role.

I believe if we had a good theory of emotional action, we would spot many other places where it was applicable. The conclusion of this survey is that certain approaches in game theory are on the right track, but need to be further developed in ways suggested by examining current research on emotions. Different techniques are necessary for different emotions. There will be no unified theory, because of the nature of the concept.
 

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