2.1. Motivations and Payoff Transformations

An important point stressed by a number of game theorists and rational choice theorists is that payoff structures of games have to capture all motivationally relevant aspects of players’ evaluations of the possible outcomes of those games.Footnote ⁷ Otherwise, without knowing precisely what games people play in terms of their true motivations, it would often be impossible to say anything about whether their choices are rational and why they make the choices that they do. To illustrate this, consider the Prisoner’s Dilemma game in terms of monetary payoffs shown in Figure 3(a). Suppose that the row player is a pure altruist when it comes to decisions involving money – he or she always strives to maximize the other player’s monetary gain. The column player, on the other hand, prefers to maximize his or her personal monetary payoff, but is averse to inequitable distributions of gains among individuals. Suppose that his or her inequity aversion is such that any unequal distribution of monetary gains is just as good in his or her eyes as personally gaining nothing. The correct representation of these players’ motivations transforms the monetary Prisoner’s Dilemma game into the game shown in Figure 3(b). There is one Nash equilibrium in the transformed game, (C, C), and the players would rationally cooperate if they both endorsed individualistic best-response reasoning and believed each other to do likewise. If we analysed observed cooperation using the original monetary payoff structure of the game, we may incorrectly conclude their choices to be irrational or presume the players to be reasoning as members of a team when that may not actually be so.

Figure 3. Monetary Prisoner’s Dilemma game (a) and its transformation (b).

This raises the question of whether it is possible to represent a shift in a player’s mode of reasoning to reasoning as a member of a team using a payoff transformation to reflect his or her motivation to attain the team’s objective in a similar way as in the case of altruism and inequity aversion above. To see that it is not, consider again the Hi-Lo game discussed earlier. Suppose that, from the point of view of a team, the outcome (Hi, Hi) is deemed to be the best, the outcome (Lo, Lo) is deemed to be the second-best and the outcomes (Hi, Lo) and (Lo, Hi) – the worst. Replacing any of the two players’ original payoffs in each cell of the game matrix with those corresponding to the team’s ranking of the four outcomes does not change the payoff structure of the original game in any way. As a result, the set of Nash equilibria and, hence, rational solutions in terms of best-response reasoning in the transformed game would be the same as in its original version, which is exactly what the theory of team reasoning was developed to contest. The key difference here is that individualistic best-response reasoning prescribes evaluating and choosing a particular strategy based on the expected personal payoff associated with that strategy, whereas team reasoning prescribes evaluating outcomes of a game from the perspective of a team (given the interacting players’ personal preferential rankings of those outcomes) and then choosing a strategy that is associated with the optimal outcome for the team. As such, the motivational shift that takes place with a switch from one mode of reasoning to another cannot be captured by transforming players’ payoff numbers in the cells of considered game matrices.Footnote ⁸

Another question is whether a shift from best-response reasoning to reasoning as a member of a team (or vice versa), in addition to changing the way a player reasons about which course of action it is best to take, may also change the way he or she personally values different outcomes of a game. In other words, whether such a shift transforms the payoff structure of a game itself. While this is an interesting possibility and one that has not been widely discussed in the literature on the theory of team reasoning to date, we assume here that this does not happen. We believe, however, there to be a reasonable ground for taking this approach. If a shift in the mode of reasoning may change the way a player personally values different outcomes, interactions between individuals could become games of incomplete information about each other’s payoffs and, as such, would take us far away from the type of games which the theory of team reasoning was developed to address. First, the solutions prescribed by best-response, team, and possible other modes of reasoning in cases of incomplete information about payoff structures of games would often be significantly different from those when payoff structures are commonly known. Players’ strategy choices would depend not only on modes of reasoning they use in a particular strategic interaction, but also on their beliefs about which payoff structures correctly define the games they play. Second, the theory of team reasoning was originally developed to resolve certain types of games in which best-response reasoning was deemed to fall short. In the case of the Hi-Lo game, for example, it was meant to resolve precisely this game with its particular payoff structure and when this structure was commonly known by interacting players. As such, for the rest of this paper we assume common knowledge of payoff structures of games and that shifts in modes of reasoning leave players’ personal payoffs associated with possible outcomes of games unchanged.Footnote ⁹

The last point to note is that reasoning as a member of a team does not imply the sharing of the attained personal payoffs among members of a team. In other words, the attained payoffs are assumed not to be transferable. If players were able to share their gains with others, such strategies and the associated payoff distributions would have to be included in representations of their strategic interactions.

2.2. Team Reasoning and Team Interests: Two Questions

The theory of team reasoning needs to address two important questions: ‘when do people reason as members of a team?’ and ‘what do people do when they reason as members of a team?’. In other words, it needs to suggest why and under what circumstances people may reason as members of a team and, once they do, what it is that they take team interests to be. In this paper we predominantly focus on the second question, though we will suggest some tentative ideas concerning the first question at the end.

3.1. Self-Sacrifice and Mutual Advantage

Bacharach (Reference Bacharach2006) mentions Pareto efficiency as a minimal condition for any function of team interests to satisfy: if one strategy profile is superior to another in terms of Pareto efficiency, a team prefers the former to the latter.Footnote ¹⁰ A function that satisfies this criterion is the maximization of the average or the sum of the interacting players’ payoffs, and it is discussed as an example in some of the early developments of the theory (e.g. Bacharach Reference Bacharach1999, Reference Bacharach2006) as well as in some more recent works (e.g. Colman et al. Reference Colman, Pulford and Rose2008, Reference Colman, Pulford and Lawrence2014; Smerilli Reference Smerilli2012). Even though it is used merely as an example, it is able to resolve some games definitively in intuitively compelling ways. It is easy to see that in the Hi-Lo and the Prisoner’s Dilemma games discussed earlier it selects the outcomes (Hi, Hi) and (C, C). One feature of this function, however, is that it may sometimes advocate a complete sacrifice of some individuals’ personal interests for the benefit of others. Consider a slight variation of the Prisoner’s Dilemma game shown in Figure 4. Here this function would prescribe the attainment of the outcome (D, C). As such, it would advocate a complete sacrifice of the column player’s personal interests for the benefit of the row player alone.

Figure 4. A variation of the Prisoner’s Dilemma game.

Although no alternative formal representations of team interests have been proposed, Sugden (Reference Sugden2011, Reference Sugden2015) discusses the notion of mutual advantage as another criterion for a function of team interests to satisfy. The idea is that an outcome selected by a team should be mutually beneficial from every team member’s perspective. Although he does not present an explicit function, Sugden proposes to define mutually advantageous outcomes as those that are associated with decision-makers’ personal payoffs satisfying a particular threshold. The suggested threshold is each player’s personal maximin payoff level in a game – the highest payoff that a player can guarantee himself or herself irrespective of what other players in the game are going to do. For both players in the Hi-Lo game this is 0. In the Prisoner’s Dilemma game of Figures 2 and 4 this is 1 (the lowest possible payoff associated with strategy D).

For Sugden, an outcome of a game is mutually beneficial if everyone’s maximin threshold is met and each player’s participation is necessary for the attainment of payoffs associated with that outcome. Notice that, according to this definition, all Nash equilibria in the Hi-Lo game are mutually beneficial. This is because every equilibrium is associated with personal payoffs higher than each player’s maximin threshold and each player’s play of the strategy associated with a particular equilibrium is necessary for the attainment of those payoffs. Thus, the above definition of mutual advantage does not exclude Pareto inefficient outcomes and, by itself, it does not suggest a further ranking of outcomes for a team once all mutually beneficial outcomes are established.

3.2. Interpersonal Comparisons of Utility

Any function of team interests that aggregates decision-makers’ payoffs requires them to be interpersonally comparable. In the Prisoner’s Dilemma game of Figure 4 this suggests, for example, that the row player prefers the outcome (D, C) to the outcome (C, C) to a greater extent than the column player prefers (C, D) to (D, D). Such comparisons go beyond the standard assumptions of expected utility theory, which make numerical representations of individuals’ preferences possible, but do not automatically grant their interpersonal comparability. This means, for example, that the representation of row player’s preferences in Figure 4 allows us to say that he or she prefers the outcome (C, C) to (D, D), or that he or she prefers the outcome (D, C) to (C, C) by a greater extent than he or she prefers (D, D) to (C, D). However, it does not allow us to claim that the row player’s well-being, in some objective sense, is the same in the outcome (C, C) as that of the column player.

It is a result of expected utility theory that any payoff function that numerically represents a decision-maker’s preferences is unique up to positive affine transformations. This means that if u is a payoff function that maps every strategy profile in a game to a real number in a way that correctly represents an individual’s preferences over outcomes, then so is any function u′ = au + c, where a > 0 and c are constants. (For a detailed discussion of why this is so, see, for example, Luce and Raiffa Reference Luce and Raiffa1957, ch. 2.) What follows from this is that the payoff structure of the Prisoner’s Dilemma game of Figure 2 represents exactly the same preferences of the interacting players as the one in Figure 5.Footnote ¹¹ As such, we would expect solutions of this game to be invariant under positive affine transformations of players’ payoffs and a function of team interests to select the same outcome(s) as optimal for a team in both cases. The maximization of the average or the sum of the two players’ payoffs, however, selects different outcomes: (C, C) in the case of Figure 2 and (D, C) in the case of Figure 5.

Figure 5. Another representation of preferences in the Prisoner’s Dilemma game.

Any version of the theory of team reasoning that uses aggregative functions of team interests will be applicable only in those contexts in which interpersonal comparisons of payoffs are possible. Since this goes beyond the standard assumptions of expected utility theory, the possibility to make such comparisons may need a separate justification.Footnote ¹² The function of team interests which we present in the next section is applicable in both cases. We will first present a version that applies to situations in which interpersonal comparisons of payoffs are not warranted. We will then explain how it can be modified to apply in cases in which payoffs are interpersonally comparable.

4.1. Formalization

For a formal presentation of the proposed function of team interests, let I = {1, . . ., m} be a finite set of m players and S _i be a set of pure strategies available to player i ∈ I. A pure strategy outcome is defined as a strategy profile s = (s ₁, . . ., s _m ), where s _i ∈ S _i is a particular pure strategy of player i. Let S = × _{i ∈ I} S _i be the set of all possible pure strategy profiles in a game, and $u_i : S \rightarrow \mathbb {R}$ be a payoff function that maps every pure strategy profile to a personal payoff for player i. A mixed strategy of player i is a probability distribution over S _i . Let Σ _i be a set of all such probability distributions and σ _i ∈ Σ _i be a particular mixed strategy of player i, where σ _i (s _i ) denotes probability assigned to s _i . A mixed strategy outcome (henceforth, outcome) is defined as a strategy profile σ = (σ₁, . . ., σ _m ). Let Σ = × _{i ∈ I} Σ _i be the set of all possible mixed strategy profiles and u _i (σ) = ∑ _{s ∈ S} (∏ _{i ∈ I} σ _i (s _i ))u _i (s) be expected payoff to player i associated with the mixed strategy profile σ.

In what follows, we present the function when mixed strategy play is allowed.Footnote ¹⁴ (For a version of the function when only pure strategies are considered simply replace σ _i , σ, Σ _i , and Σ with s _i , s, S _i , and S.) Let u ^max _i ≔ max_{σ ∈ Σ} u_i (σ) denote player i’s personal payoff associated with his or her most preferred outcome, let u ^ref _i denote i’s reference payoff from which individual advantage to i is measured, and let u ^maximin _i ≔ max_{σ i ∈ Σ i} {min_{σ−i ∈ Σ−i} u_i (σ)} denote i’s maximin payoff level in the game (where σ_−i ∈ Σ_−i denotes a combination of strategies of all players other than i).

So long as u ^ref _i ≠ u_i ^max, the extent of individual advantage to player i associated with a particular strategy profile σ is

(1) $$\begin{equation} u_i^\iota (\sigma ) = \frac{u_i(\sigma )-u_i^{\rm {ref}}}{u_i^{\rm {max}}-u_i^{\rm {ref}}} \end{equation}$$

(Notice that if i’s payoff function u _i is normalized so that u ^max _i = 1 and u ^ref _i = 0, then u ^ι _i (σ) = u_i (σ).) Having defined u ^ι _i (σ), the extent of mutual advantage associated with σ is

(2) $$\begin{equation} u^{\tau} (\sigma ) = {\min\limits_{{i \in I}}}\, u_{i}^{\iota} (\sigma ) \end{equation}$$

When u ^ref _i ≠ u_i ^max for all i ∈ I, the proposed function of team interests $\tau : \mathcal {P}(\Sigma ) \rightarrow \mathcal {P}(\Sigma )$ , where τ(Σ) = Σ^τ⊆Σ, selects a subset from the set of all possible strategy profiles in a game such that each selected strategy profile maximizes the extent of mutual advantage to the interacting players, subject to each player’s personal payoff being at least as high as his or her maximin payoff level in the game. Formally, each element σ ∈ Σ^τ is such that

(3) $$\begin{eqnarray} \begin{array}{rl} \sigma \in &\ {\arg \max\limits_{\sigma \in \Sigma }}\, u^\tau (\sigma ) \\ &\ \mbox{subject to}\ \forall i \in I : u_i(\sigma ) \ge u_i^{\rm {maximin}} \end{array} \end{eqnarray}$$

or, inserting (1) and (2) into (3),

(4) $$\begin{eqnarray} \begin{array}{rl} \sigma \in &\ {\arg \max\limits_{\sigma \in \Sigma }} \left\{ {\min\limits_{i \in I} }\, \displaystyle\frac{u_i(\sigma )-u_i^{\rm {ref}}}{u_i^{\rm {max}}-u_i^{\rm {ref}}} \right\} \\ &\ \mbox{subject to}\ \forall i \in I : u_i(\sigma ) \ge u_i^{\rm {maximin}} \end{array} \end{eqnarray}$$

If u ^ref _i = u_i ^max for some i ∈ I, u ^ι _i is undefined and nothing is individually advantageous to player i. Consequently, nothing is mutually advantageous to the group of players I and so τ(Σ) = ∅. To summarize,

(5) $$\begin{equation} \tau (\Sigma ) = \left\lbrace \begin{array}{ll}\Sigma ^\tau & \mbox{when $u_i^{\rm {ref}} \ne u_i^{\rm {max}}$ for all $i \in I$} \\ \emptyset & \mbox{otherwise} \end{array} \right. \end{equation}$$

4.2. Applicability and Two Properties

The application of the above function of team interests is limited to cases in which u ^ref _i and u ^max _i can be identified for every payer i in a game. A sufficient condition for this to be so is having a finite number of players and pure strategies available to each player. With an infinite number of players and/or pure strategies this may not be so. Consider, for example, a decision situation in which a number of firms compete to employ a prospective employee by offering a salary. Since there is no upper bound on the salary a firm could offer in theory, there is no upper bound on the payoff the potential employee may attain.Footnote ¹⁵ In this and similar scenarios our proposed function of team interests could only be applied if all players’ payoff functions were bounded from above and below. There may be cases, however, in which all players’ payoff functions are bounded, but the infinite number of strategies allow one’s payoff to approach a particular value, but never quite reach it. Suppose that the aforementioned competing firms are prohibited from paying anyone more than some set salary level. If infinitesimal fractions of monetary units were meaningful, theoretically this would mean than the maximum payoff for the prospective employee could not be identified. In such and similar scenarios our discussed function could only be applied by approximating each player’s u ^ref _i and u ^max _i payoffs.

Two properties of τ can be derived without further specification of u ^ref _i . First, provided there is a finite number of players and pure strategies available to each player, and u ^ref _i ≠ u_i ^max for all i ∈ I, the set of strategy profiles selected by τ is nonempty. This is so because, for every player i in a game, there always exists at least one maximin strategy $\sigma _i^{\rm {maximin}} \in \arg \max _{\sigma _i \in \Sigma _i} \lbrace \min _{\sigma _{-i} \in \Sigma _{-i}} u_i(\sigma )\rbrace$ , such that u_i (σ^maximin _i , σ_−i ) ⩾ u ^maximin _i . As such, there is at least one strategy profile σ^maximin = (σ^maximin ₁, . . ., σ _m ^maximin), such that u_i (σ^maximin) ⩾ u ^maximin _i for every player i ∈ I, which satisfies the constraint in (3) and (4). Since the function τ selects strategy profiles associated with maximum mutual advantage from those that satisfy the above constraint and since there is at least one strategy profile that satisfies it, it follows that Σ^τ is nonempty.

Another property of τ is that every strategy profile that it selects is efficient in the weak sense of Pareto efficiency.Footnote ¹⁶ To see this, suppose that τ selects the strategy profile σ ^x ∈ Σ when there exists another strategy profile σ ^y ∈ Σ, such that u _i (σ ^y ) > u _i (σ ^x ) for every player i ∈ I (in other words, σ ^x is not Pareto efficient in the weak sense). As long as u ^ref _i ≠ u_i ^max for every i ∈ I, it follows that min _{i ∈ I} u ^ι _i (σ ^y ) > min _{i ∈ I} u ^ι _i (σ ^x ). Hence, $\sigma ^x \notin \arg \max _{\sigma \in \Sigma } \lbrace \min _{i \in I} u_i^\iota (\sigma ) \rbrace$ , and so σ ^x ∉Σ^τ.

4.3. Reference Points

The above function of team interests can be applied with any set of personal reference points, relative to which individual and, based on these, mutual advantages are measured. We discuss three possible reference points here and focus on one of them in more detail when reviewing examples in the next section.

Since every outcome of a game is reachable via players’ joint actions, one possibility is to set each player’s reference point to be the worst payoff that he or she can attain in that game. It is the payoff associated with a player’s least preferred outcome: u ^ref _i = min_{σ ∈ Σ} u_i (σ). However, this approach may be criticized on the basis that not all outcomes of games should be considered on an equal footing when it comes to establishing players’ personal reference points, relative to which individual advantages are to be measured. It may be argued that outcomes that are not rationalizable in terms of best-response reasoning should be left out from the set considered for this purpose. Since this would exclude strictly dominated strategies, outcomes defined in terms of such strategies would not be used for establishing reference points.Footnote ¹⁷ Letting Σ ^br ⊆Σ denote the set of strategy profiles that are rationalizable in terms of best-response reasoning, this modifies the previous reference point to the following: u ^ref _i = min_{σ ∈ Σ ^br} u_i (σ).

The fact that the notion of rationalizability is associated with best-response reasoning raises the question of whether it should be applied in establishing reference points for identifying mutually advantageous outcomes when using a mode of reasoning that is not based on best-response considerations. There are two ways to justify the use of the restricted set Σ ^br instead of Σ. The first is to argue that decision-makers approach their considered games in terms of best-response reasoning to begin with, and that, having drawn conclusions about outcomes they could potentially reach through the application of this mode of reasoning, they evaluate everything that follows – including their considerations about what is mutually advantageous – relative to those conclusions. We will briefly return to this idea later in Section 5. The second approach is to argue that players’ reference points, relative to which individual advantages are measured to subsequently establish the extents of mutual advantage, are, essentially, individualistic. These are thresholds, such that anything that is preferentially inferior to them is not individually advantageous to respective decision-makers. Hence it may be argued that their establishment should be based on individualistic reasoning and, as such, best-response rationalization is a plausible restriction to impose.

The third possible reference point is a player’s personal maximin payoff level in a game: u ^ref _i = max_{σ i ∈ Σ i} {min_{σ−i ∈ Σ−i} u_i (σ)}. This approach is closest to the definition of mutual advantage discussed by Sugden (Reference Sugden2015) and it ensures that the maximin constraint in the function τ is automatically met. However, the previous criticism could still apply: strategy profiles associated with players’ maximin payoffs may be excluded from the set of outcomes that are rationalizable in terms of best-response reasoning. Also, outcomes associated with these thresholds, when they lie close, in relative preferential terms, to players’ most preferred outcomes of games, may, at times, serve as useful definitive resolutions of those games when individualistic best-response reasoning leads to indeterminacies. As such, maximin thresholds themselves may be mutually advantageous. We will illustrate such a case shortly.

4.4. Examples

In this section we show what the function τ selects in a few simple examples. Two of these – the Hi-Lo and the Prisoner’s Dilemma games – we already introduced. The other two are a version of the Chicken game and a game we labelled the High Maximin shown in Figure 6. In our discussion here we compute the extents of individual and mutual advantage using the second of the three possible reference points discussed in the previous section: u ^ref _i = min_{σ ∈ Σ ^br} u_i (σ). We present a detailed derivation of results in the Chicken game first and summarize the function’s prescriptions for the remaining games afterwards.

Figure 6. The Chicken (a) and the High Maximin (b) games.

There are three Nash equilibria in the presented version of the Chicken game: (U, L), (D, R), and (U $\frac{6}{7}$ , D $\frac{1}{7}$ ; L $\frac{1}{7}$ , R $\frac{6}{7}$ ). The third is a mixed strategy equilibrium, in which row player randomizes between U and D with probabilities 6/7 and 1/7, while column player randomizes between L and R with probabilities 1/7 and 6/7 respectively. This yields both players an expected payoff of 10/7. Any combination of strategies associated with the three Nash equilibria is rationalizable in terms of best-response reasoning. As such, for both players, the least preferred rationalizable outcome from the set Σ ^br is (U, R), the maximin payoff level is 1 (the lowest possible payoff associated with strategies D and L), and the most preferred outcome yields a payoff of 10. Thus, for both players, u ^ref _i = 0, u ^max _i = 10, and u ^maximin _i = 1. The extents of individual and mutual advantage, u ^ι _i and u ^τ, associated with the four pure strategy outcomes of the game are shown below (sorted by u ^τ). These, as noted earlier, are expressed in percentage terms (u ^ι _i and u ^τ are multiplied by a factor of 100).

When only pure strategies are considered the maximally mutually advantageous outcome is (D, L) and, since it satisfies the maximin constraint for both players, it is the unique outcome selected by τ. The result is slightly different, albeit also yielding a unique solution, if mixed strategies are considered as well. In the latter case, maximum mutual advantage is associated with the mixed strategy profile (U $\frac{3}{14}$ , D $\frac{11}{14}$ ; L $\frac{11}{14}$ , R $\frac{3}{14}$ ), which yields both players an expected payoff (with reference to the payoff structure in Figure 6(a)) of approximately 4.32. The corresponding approximate extent of mutual advantage is 43.2, which is higher than that associated with (D, L). As a result, S ^τ = {(D, L)} when only pure strategies are considered and $\Sigma ^\tau = \lbrace \mbox{({\em U} $\frac{3}{14}$, {\em D} $\frac{11}{14}$; {\em L} $\frac{11}{14}$, {\em R} $\frac{3}{14}$)}\rbrace$ when mixed strategies are considered as well. Either way, S ^τ and Σ^τ are singletons, which means that the function τ resolves this game definitively for those who reason as members of a team. Also note that in both cases τ selects a non-Nash-equilibrium outcome.

Results for the remaining three games are summarized in Table 1. The top section of the table shows parameter values and the selected outcomes in S ^τ when only pure strategies are considered (u ^maximin _i is abbreviated for row and column player as u ^mxm _row and u ^mxm _col respectively). The bottom section shows these values and the selected outcomes in Σ^τ when mixed strategies are considered as well.

Table 1. Summary of results in four examples.

*In Σ^τ for the High Maximin, 0 ⩽ p ⩽ 1/10.

In the case of mixed strategies in the Hi-Lo game, the maximin strategy for both players is to randomize between Hi and Lo as in the mixed strategy Nash equilibrium. Irrespective of whether mixed strategies are considered or not, however, (Hi, Hi) is the unique outcome selected by τ.

The three Nash equilibria in the High Maximin game are (U, L), (D, R), and (U $\frac{10}{11}$ , D $\frac{1}{11}$ ; L $\frac{9}{10}$ , R $\frac{1}{10}$ ). The mixed strategy equilibrium yields expected payoff of 9 and 10/11 to row and column player respectively. In the case of mixed strategies, column player can secure an expected payoff of at least 10/11 by randomizing between L and R with probabilities 10/11 and 1/11. As in the Chicken game, the output of τ depends on whether mixed strategies are considered or not. If they are, any strategy profile in which the row player plays D while the column player randomizes between L and R with probabilities 0 ⩽ p ⩽ 1/10 and 1 − p respectively is maximally mutually advantageous and is included in the set Σ^τ, which means that Σ^τ is not a singleton. This still resolves the game definitively for the interacting players, since τ prescribes a unique strategy choice to row player and it is up to column player alone to select any outcome from the set Σ^τ using 0 ⩽ p ⩽ 1/10 of his or her choice. If only pure strategies are considered, τ selects (D, R). Note that mutually advantageous play in both cases yields row player his or her maximin payoff. As such, this is an example of a case where a personal payoff associated with maximally mutually advantageous outcome(s) is also one player’s maximin level. By contrast, if u ^ref _i were set for both players to their personal maximin payoffs, τ would select the outcome (U, L) with and without mixed strategy play.

Lastly, in the Prisoner’s Dilemma game the function τ selects (C, C). Parameter values in Table 1 are based on the Prisoner’s Dilemma game of Figure 2, but the result is the same for all versions of this game discussed earlier.

4.5. No Irrelevant Player or Strategy

A point to note is that the function τ does not ignore any player in its determination of team-optimal outcomes. This means that no player can be dismissed on grounds of being irrelevant. One consequence of this is that the presence of a player who is indifferent between all outcomes of a game, with u ^ref _i = u_i ^max, renders Σ^τ = ∅. Since such a player personally does not care about how the game is going to be resolved, there is nothing that would motivate him or her to seek out mutually advantageous solutions. Although in cases involving more than two players this leaves the possibility for the remaining decision-makers to consider what is mutually advantageous for them based on their predicted actions of those outside of their group, mutually advantageous play is not possible for all players in the game as members of one team.

Also, as the Prisoner’s Dilemma example shows, the addition of strictly dominated strategies to a game can result in changes in the set of outcomes selected for a team.Footnote ¹⁸ This means that such strategies cannot be treated as irrelevant either. With regards to this point, however, the function τ can be adapted to a more orthodox interpretation of what counts as feasible in games by limiting its output to outcomes that are rationalizable in terms of best-response reasoning. For this approach, all outcomes outside of the set Σ ^br are excluded from those considered as potential outputs of τ, and u ^ref _i with u ^max _i are assigned payoff values associated with each player’s least and most preferred outcome(s) in Σ ^br . Returning to our examples, in the Hi-Lo game this would yield the same results as previously. In the Chicken and the High Maximin games the outcomes selected by τ when mixed strategies are considered would be the same as those selected for pure strategy play earlier (this is because previously selected mixed strategy profiles are not rationalizable in terms of best-response reasoning and, as such, they would be excluded). Lastly, in the Prisoner’s Dilemma game nothing would count as mutually advantageous, since only one outcome is rationalizable in terms of best-response reasoning. This version of τ would allow to regard all strictly dominated strategies as irrelevant.

4.6. Interpersonal Comparisons of Advantage

Earlier we noted that the standard expected utility theory makes numerical representations of decision-makers’ preferences possible, but does not automatically grant their interpersonal comparability, and suggested that the function τ applies in cases when interpersonal comparisons of payoffs are not warranted in this sense. The presented function, however, does make an interpersonal comparison of one sort: it equates a unit of individual advantage gained by one player in a game with that gained by another. We argue that even when the comparison of the former sort is not warranted, the comparison of the latter sort is nevertheless permissible.

Interpersonal comparisons of decision-makers’ payoffs tend to associate their preference satisfaction with some interpersonally comparable notion of attained well-being or other form of welfare. If, in the context of games, we were to contemplate the possibility that for one player stakes might be significantly higher than they are for another in this sense, we would indeed be making an implicit assumption that their attained payoffs can be interpersonally compared. Our proposed measures of individual and mutual advantage, expressed as extents of relative advancement of players’ personal interests towards their most preferred outcomes of games, however, are meant to be applied when such comparisons are not possible (or, to put differently, when such comparisons provide no meaningful information). The reason they can be applied is the fact that the scales for these measures can be established using commonly known and objectively identifiable points in games without the need to make interpersonal comparisons of the attained payoffs. In order to use these measures, decision-makers need to know each other’s preferences over the possible outcomes of games and their personal reference points, but they need not be able to make any interpersonal comparisons of their attained well-being in some objectively or subjectively comparable way. Without comparing their attained well-being players can still compare the extents to which their personal interests are advanced within the context of some particular decision problem they are trying to solve.Footnote ¹⁹

The interpersonal comparisons implied by the function τ are thus quite different from those not warranted by the standard expected utility theory. When decision-makers consider mutually advantageous play in games, they equate units of measures of their individual advantage – the advancement of their personal interests relative to what they personally deem to be the best and the worst case outcomes of their interactions – while not being able to equate units of the attained personal well-being. All they know is how much a particular outcome is individually advantageous to a player relative to that player’s reference point and his or her most preferred outcome of a game.Footnote ²⁰

This does not mean that interpersonal comparisons of payoffs are never possible or that we never make them in our interactions with each other. Binmore (Reference Binmore2005, Reference Binmore, Ross and Kincaid2009) argues, for example, that we may have evolved the ability to make such comparisons over the course of our evolutionary development. This, however, does not preclude the applicability of the above function for identifying mutually advantageous outcomes. A step that needs to be added before the function is applied in cases when players’ payoffs are interpersonally comparable is the rescaling of players’ measures of individual advantage u ^ι _i using appropriate scaling factors to equate these units in accordance with how their attained personal payoffs interpersonally compare.

4.7. Mutual Advantage and Coordination

There will be many games in which the function τ selects more than one outcome. In some situations, such as the High Maximin game of Figure 6(b), this will not cause a difficulty for decision-makers to coordinate their actions, but in many cases it will. We may thus consider whether the expected success of coordinating actions on some particular outcome should be reflected in the measure of mutual advantage associated with that outcome.

Consider the expanded version of the Hi-Lo game shown in Figure 7(a). There are seven Nash equilibria in this game, three in pure strategies and four in mixed. (The four mixed strategy equilibria yield each player an expected payoff of at most 5.) The function τ selects two outcomes as maximally mutually advantageous: (Hi1, Hi1) and (Hi2, Hi2). As such, it leaves team-reasoning decision-makers facing a coordination problem. Since the selected outcomes are indistinguishable (apart from strategy labels and their positions in the matrix representation of the game, both of which can aid to coordinate players’ actions – a point which we ignore for the time being), if the two players, without communicating with each other, were to simultaneously attempt to attain any one of the two, they could expect to succeed with probability 1/2. This would yield each player an expected payoff of 5. The outcome (Lo, Lo), however, is unique in the sense that it is the only outcome of the game that yields both players a payoff of 9. It may thus be beneficial for the interacting players, instead of attempting to coordinate their actions on one of the two indistinguishable outcomes, to focus on the attainment of the outcome (Lo, Lo). This would guarantee each player a certain personal payoff of 9 instead of the expected payoff of 5.

Figure 7. The expanded Hi-Lo game (a) and its variations (b), (c).

Bardsley et al. (Reference Bardsley, Mehta, Starmer and Sugden2010) and Faillo et al. (Reference Faillo, Smerilli and Sugden2016) discuss the idea of incorporating perceived coordination success rates when it comes to evaluating the attainment of one from a number of indistinguishable outcomes of a game into the function of team interests itself and use it to interpret results from a number of experiments.Footnote ²¹ It is easy to modify the measure of mutual advantage u ^τ to do this. In a case of a coordination problem due to multiplicity of outcomes that are indistinguishable from a team’s perspective and in the absence of any further coordination aid for the interacting players to rely on, the original measures of mutual advantage associated with the indistinguishable outcomes can be divided by the number of indistinguishable outcomes in question. In the game of Figure 7(a), this would yield the extent of mutual advantage associated with outcomes (Hi1, Hi1) and (Hi2, Hi2) to be 50, while that associated with the outcome (Lo, Lo) would remain to be 90. Operationalized this way, the function τ would now select the outcome (Lo, Lo) as uniquely optimal for a team. However, although payoff pairs associated with the pure strategy Nash equilibria in this game can themselves aid to coordinate team-reasoning decision-makers’ actions in this way, coordination of actions may often be possible due to factors that are not related to payoffs associated with outcomes that are deemed indistinguishable from a team’s perspective. For one example, consider a variation of this game shown in Figure 7(b). The pure strategy Nash equilibria here are the same as earlier and the function τ, based on the original measure u ^τ, selects the same pair of outcomes, (Hi1, Hi1) and (Hi2, Hi2), as optimal for a team. Although the two outcomes are just as indistinguishable as they were before, in this case the personal payoff of 8 to row and column player in the outcome (Hi2, Hi1) and (Hi1, Hi2) off the matrix diagonal respectively can aid to coordinate players’ actions for the attainment of the outcome (Hi2, Hi2). Even though it is still the payoff structure of the game that aids players to resolve their coordination problem in this case, it is not a payoff pair associated with any of the outcomes on which the interacting players try to coordinate their actions that does so.

Another example is shown in Figure 7(c). Here everything is the same as in the original version of the game, except for the fact that the outcome (Hi1, Hi1) is marked with a star. The salience of this outcome has nothing to do with the payoff structure of the game, but the presence of the star serves as a potential aid to coordinate actions. In fact, there is a number of coordination aids to choose from: (i) the presence of the star among outcomes that are maximally mutually advantageous in terms of the original measure u ^τ, (ii) its absence, and (iii) the uniqueness of the payoff pair associated with the Nash equilibrium (Lo, Lo). Which of these is most appropriate will depend on decision-makers’ beliefs about which of these aids is most likely to be recognized and considered by others.

Our intention with these examples is to show that decision-makers’ ability to coordinate their actions on a particular outcome of a game may often depend on factors that have nothing to do with how mutually advantageous it would be for the players to end up at that outcome in terms of their personal payoffs associated with it. Furthermore, the possibility of there being multiple coordination aids for players to choose from may leave them facing a coordination problem of a different kind – one related to the choice of a coordination aid to resolve the game they play. The interacting players’ choice of the latter will depend on their beliefs about which aids are most likely to be recognized, considered, and adopted by other players. These beliefs may, in turn, depend on decision-makers’ cultural backgrounds, the prevalent social norms in their societies, and other factors unrelated to the payoff structure of the game itself.Footnote ²² Because of this it may often be fruitful to keep the question of which outcomes of games are maximally mutually advantageous in terms of players’ attainable payoffs separate from the question of how to successfully coordinate their actions. Keeping the two questions separate allows the latter to be informed by the former: in the presence of many potential coordination aids to choose from, the measure u ^τ may help narrow down which of these should be employed in order to remain as faithful as possible to the mutual advancement of players’ personal interests. In this light, it is the fact that the outcome (Hi, Hi) in the original version of the Hi-Lo game of Figure 1 is maximally mutually advantageous to the interacting players in terms of the attainable personal payoffs that makes them coordinate their actions on this outcome rather than the less mutually advantageous one, (Lo, Lo).