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Players in economic situations often have preferences not only over their own outcome but also over what happens to fellow players, entirely apart from any strategic considerations. While this can be modeled directly by simply writing down final preferences, these are commonly unknown

Frank [

This paper attempts to provide a general, formal, theoretical link between the base payoffs in a game, and the resulting final utilities or preferences. The discrepancy is due to the fact that players care about the utilities of the other players in the game, e.g. due to altruism. The main reason to formalize this link is to provide applied and experimental economists with a model for this pervasive interaction, so they are not forced to come up with new (and ad hoc) formulations every time it is relevant. There is also a second reason, the stock-in-trade of theorists: to understand the process better. The jump from payoffs to final utilities goes on all the time in almost all games, so we should have a model (or, better yet, several competing models) of how it happens and what it implies.

We introduce a general definition of games with synergistic utility. Synergistic utility functions capture the idea that utility increases in one’s own payoff, and may increase or decrease in others’ utilities. Sufficient technical conditions are imposed for the concept to be well-defined, but otherwise the formulation is general enough to allow maximal variety in specific applications. All players are fully rational (including being expected-utility maximizers) and no new equilibrium concepts are introduced. A specific example, the linear synergistic utility function, is introduced and analyzed in greater detail. Several applications of the theory are given, including: cooperation in the Prisoner’s Dilemma, overproduction in Cournot oligopoly, extended play in the centipede game, and interior solutions in the dictator game.

The paper proceeds to

The literature relating to altruism and interdependent preferences is wide and diverse, with each paper seemingly taking its own course. The first broad category can be considered to be the various applications of altruistic-like tendencies in specific situations. This includes, in the macroeconomics literature using overlapping generations (OLG) models, the famous paper of Barro [

Altruism within the family has been studied since Becker [

The second general class of papers are those on evolution and biology, which are also closely tied to the theoretical psychology literature. Frank [

A large number of experimental economics papers have looked at different games and found results that diverge from those predicted by the basic equilibrium concepts. Dawes and Thaler [

This leads naturally to the final group of related papers, those from the game theory literature. Geanakoplos, Pearce and Stacchetti [

Returning to the traditional equilibrium concepts, Bergstrom [

One way to introduce an altruism-like aspect in a formal game-theoretic model is to add a positive constant to payoffs following a “good” action, such as contributing in a public goods game or cooperating in the Prisoner’s Dilemma. This “warm glow” effect is plausible in some circumstances, but does not capture the positive or negative benefits that a player may receive depending on the welfare of his or her opponents

It is not unreasonable to ask why utilities should not be a function of own

One final matter that should be clarified before proceeding to the formal model is the interpretation of the base payoffs. They are already objects in utility space, so they should not be thought of as monetary payoffs or profits. Rather, they can be considered to be the utility resulting from that outcome if it were in a one-person setting, or in a setting where the effects of that outcome on other players are unknown. Alternately, they are the utilities of thoughtless players, to whom it has not yet occurred that there are other players or what implications that might entail. We assume, as ever, that they already include any positive feelings from simply doing good or being fair, or on the flip side any negative feelings directly arising from an act of, say, betrayal. What they do not include are preference changes due to the realized utility of one’s opponents in a particular outcome of the game

We are given a game _{i}_{i}^{I}

^{I}

(i)

(ii)

(iii) there exists _{E} ∈ Θ such that for all vectors ^{I}_{E}) = _{1}

(iv) for all

(v) for all ^{I}^{∞} (

In words, then, requirement (i) states that utility must be increasing in one’s own payoff. Requirement (ii) asks that utility, if it is affected by someone else’s payoff, always be affected in the same direction. This could be weakened, but imposes no untoward restrictions

_{i}

_{i}_{i}_{-i}_{i}

The proposition says that the limit utilities, which necessarily exist, satisfy a fixed-point property. The proof follows straightforwardly from the definitions and the continuity of _{i}_{-i}_{1} = _{2} = 1 then the limit diverges, as would be expected (utilities go to infinity as each player gets happier and happier contemplating the situation). The fixed-point solution, on the other hand, yields _{1} = _{2} = -1, which appears unreasonable. Thus the limit is central to the definition, but Proposition 1 may provide a short-cut in explicit calculations.

_{i}

^{I}^{I}

Proposition 2 gives us another general property of synergistic utility functions, but this is about as much as can be said in complete generality. It may be helpful at this point, in part to clarify the definitions, to consider some examples of potential synergistic utility functions. We say potential because for the moment we ignore condition (v), and we leave Θ unspecified. The most obvious is probably the linear formulation ^{I}_{i}_{‒i}^{2} is impermissible, for instance, because it violates (ii). The effect of an increase in the other player’s utility on one’s own should be independent of the absolute levels involved. Thus, _{i}_{‒i}^{3} is once again acceptable. Cobb-Douglas formulations, more common in macroeconomics, look like _{i}^{a}_{‒i}^{b}

To apply the theory in a specific situation, one must choose an appropriate (

_{i}_{‒i}_{i}_{‒i}

We may write this as ^{n}^{n‒1}, and hence ^{n}^{n}u^{0}, where ^{0}= [_{1} _{2} 1]´. Then multiplying out the powers of

But by assumption |_{1}_{2}| ˂ 1 so

Now ^{th} row of the 3^{rd} column of the matrix above so it too exists (and in fact this gives an explicit formula for it). Naturally, this is the same solution one would find from solving the system of two fixed-point equations. It is clear that conditions (i)-(iv) also hold.

Note that the perverse example mentioned earlier, which had

(a) all player’s payoffs are multiplied by the same positive constant, or

(b) any or all players have a constant added to their payoffs

(a) Since _{i} (or in fact more generally whenever _{i}), utilities all along the limiting sequence, and hence also final utilities, will also be multiplied by this constant. So then, by the standard result, preferences remain the same.

(b) Adding a constant to one player’s payoffs affects all players, but only to the extent of adding some constant to each of their payoffs. Although this constant may be different for each player, it is the same for a given player across his or her outcomes. This is clear from the explicit formulas found in the proof of Proposition 3. But now, once again, the standard result applies.

Although this result does not hold in general for all synergistic games, it will hold in other particular settings. We now turn our attention to illustrating the theory with a spectrum of examples.

The proof of the pudding lies in the taste, and the believability of synergistic utilities lies in its potential applications. For the time being, we confine ourselves to the linear synergistic utility function analyzed above, _{i}_{‒i}_{E} = (1,0)_{.} This type always has final utility equal to base payoff regardless of the other players. The second type is an altruist, denoted by

The basic Prisoner’s Dilemma can be written as:

Here

The unique NE is again for both players to defect. What is interesting, however, is that this outcome is no longer Pareto inefficient, as it was previously. The economist is so unhappy that it makes the Jones player happy. This depends, of course, on the exact payoff structure and type of player 2, but holds over a wide class. Consider next a socialist player 1 against a Jones type:

This game now has two pure NE, in both of which type

Cooperation is a dominant strategy here for both players; it is also the optimal outcome in the game. This is the stereotype of altruistic cooperation in the Prisoner’s Dilemma. The final combination of players that we consider is when player 1 is a type

The unique and strict NE is (

Turning next to an example of a continuous game, we consider Cournot duopoly. In the simplest case with linear unit demand and zero marginal cost, price _{i}_{i}

Experimental game theory has included extensive work not only with the Prisoner’s Dilemma but also with other games such as ultimatum, dictator, centipede, and public goods games. As in the case of the Prisoner’s Dilemma, the results are often quite disparate from those predicted by standard theories. For instance, no positive quantity should ever be rejected in an ultimatum game, yet this is often observed in experiments. This outcome can be explained using synergistic utilities: types similar to Jones will reject all offers up to some level (which will depend on the exact type chosen and on the type of the opponent). Of course altruism alone, without some sort of negative analogue, can never rationalize these rejections. Recall that it is possible to extend the theory to include reciprocity if desired, so a player’s type need not be constant. As has been documented previously (see

In the so-called dictator game, player one simply decides how to divide an amount of money (typically around $10 in experiments) between him- or herself and an often anonymous opponent. Player two has no action other than to accept the split as dictated. Traditional equilibrium concepts predict that player one should keep the entire amount, and previous models of altruism have not altered this prediction. For instance, continuing with the types as defined above, if an altruistic type _{2} = _{2}. For player one, we assume the altruistic formulation

Despite the fact that the game structure remains the same in synergistic games (only the payoffs have changed), there are several topics that take on new meaning in this context. For instance, cooperative games with transferable utility will be difficult to analyze since some players may actually prefer a smaller total surplus to divide (think of the type

A related consideration, though more in the mode of full rationality, is the idea of segregation. Since players are of different types, they may prefer to play against one type of opponent rather than another, and thus selectively associate. Of course, they may not have the opportunity to make this choice, but if they do then it has long-term welfare (and hence possibly evolutionary) implications. Returning once again to the Prisoner’s Dilemma example of the previous Section, note that while types

There is no doubt at least some element of reciprocity in almost all human interactions. Synergistic utilities, as defined, make no account for this; a player’s degree of altruism is independent of the attitudes of the other players. The work of Rabin [

As a first step toward examining how important reciprocity is in influencing other-regarding behavior, and as an experimental exploration of synergistic utilities, the following study could be implemented ^{0}^{1}^{2}

Finally, games with incomplete information take on an added dimension if there is also the possibility of synergistic types. There is no reason in general to assume that all players know the type of each of their opponents, synergistic or otherwise. Fortunately, the entire game-theoretic apparatus developed to analyze this eventuality is still perfectly applicable. In particular, the Bayesian equilibrium concepts apply just as well here. As synergistic types are certainly payoff relevant, signaling will be an important component to playing extensive-form synergistic games. It may or may not be beneficial for a player in a given situation to reveal his or her synergistic type (consider, for instance, the discussion of segregation above). In fact, incomplete information aspects of synergistic games seem to be perhaps the most fruitful line for future theoretical research using this model.

Game theorists assume that the payoffs in a game indicate true preferences, which is to say that they already take into account welfare interactions between the players. But often in real-life situations, the only information available is about base payoffs, e.g. profits for firms or monetary payoffs in an experimental setting. It is useful to have a specific model of altruism and other emotional aspects in order to link these payoffs to the ultimate utilities in a game. The concept of synergistic utilities attempts this, by providing a simple framework in which to address these concerns in various applied contexts. Each player’s utility is a function of his or her own payoff and of the other players’ utilities. Standard equilibrium concepts are sufficient, and since the process is a transformation of payoffs only, the theory can be applied to arbitrary games, with any number of players. One special case, a linear formulation, was given and analyzed in more detail. Examples, such as how both cooperation in the Prisoner’s Dilemma and positive gifts in the dictator game can be rationalized, followed.

The main distinction between the present work and previous literature lies in the simplicity of synergistic games. There is nothing new imposed on the game structure or analysis, since the only change made is in the numerical values of the payoffs. Nor is an idea of reciprocity inherent or necessary to the model. Nevertheless, many observed behaviors can be explained within this paradigm. Note in particular that standard theories have done exceptionally well in predicting behavior in market situations. In these games, by definition, a player cannot influence the payoff of anyone else in the game (or at least is of this impression). Hence a player with synergistic utility will behave exactly as a standard player would, a robustness check on the theory. Surely there will be more such checks to come.

Throughout the paper “opponent” will be used interchangeably with “other player”, whether or not the particular relationship happens to be adversarial.

One caveat is that this may not apply as fully in a corporate setting.

The author has worked considerably with this alternate model and is more than willing to share the results of these pursuits with anyone who is interested.

Note that we are assuming, as we must, the possibility of interpersonal comparison of cardinal utility.

Note, however, that it does not allow sufficient flexibility for very much reciprocity. This is by design: we see how much can be accomplished in as simple a setting as possible.

In general, of course, there may be several fixed-point solutions, while there is necessarily at most one limit point. This is another reason to choose the limit definition, although in synergistic games as defined multiplicity won’t be a problem.

Note that since

Note that we cannot then independently choose the cardinalization for taking expected utilities.

For example, we might imagine that more generally one would require the derivative of

Most of the previous literature has instead chosen (in its own context)

A similar Jones type appears in the macroeconomics consumption literature, so this is conceivably an example of micro keeping up with the macro Joneses.

Contrast this once again with the quote from Rotemberg (1994) in

Thanks to a referee for suggesting this line of reasoning.