2 $\xd7$ 2 games (such as the Prisoner’s Dilemma) are economic games for studying cooperation and social decision-making. Linearly-additive decomposed games are variants of 2 $\xd7$ 2 games that can change the framing of the game and thereby provide researchers with additional flexibility for measuring preferences and social cognition that would not be possible with standard (matrix-form) 2 $\xd7$ 2 games. In this paper, we provide a systematic overview of linearly-additive decomposed 2 $\xd7$ 2 games. We show which 2 $\xd7$ 2 games can be decomposed in a linearly-additive way and how to calculate possible decompositions for a given game. We close by suggesting for which experiments decomposed games might be more conducive than matrix games.

## 1. Introduction

In this article, we explain what linearly-additive decomposed 2 $\xd7$ 2 games are, how they work, and for which types of experiments they might be most useful. In our opinion, they are a versatile but underused tool for studying social preferences and social interactions. The purpose of this article is to help readers understand these games better, such that they can use them in their own research. For readability, we will omit the phrase ‘linearly additive’ (such that we refer to linearly-additive decomposed 2 $\xd7$ 2 games as ‘decomposed games’) and we put all relevant mathematics in the Appendix.

### 1.1. Social interactions and 2 × 2 games

Cooperation and social decisions have been studied for decades across a wide range of disciplines (e.g., Fudenberg & Tirole, 1991; Henrich et al., 2001; King-Casas & Chiu, 2012; Nowak, 2006; Perc et al., 2017; Rilling & Sanfey, 2011; Van Lange et al., 2013). To tease apart various aspects of cooperation, different tasks have been developed (Thielmann et al., 2021), such as the Dictator Game (Engel, 2011), the Ultimatum Game (Güth & Kocher, 2014), and the Trust Game (Johnson & Mislin, 2011).

One of the most widely studied economics games is the Prisoner’s Dilemma, which models conflict between individual and collective. In the original framing of the Prisoner’s Dilemma (see Poundstone, 1993), a police officer arrests 2 criminals. The evidence is not sufficient to convict them of the major crime, but there is evidence to convict them of a lesser crime. The officer makes both criminals, who are kept in separate rooms and cannot communicate with each other, the same offer: if both talk to the police and betray each other, they each go to prison for 5 years; if both remain silent, they each go to prison for 2 years; if one remains silent and the other speaks, the first will go to prison for 10 and the latter will be set free (see Figure 1).

The Prisoner’s Dilemma thus models a conflict between what’s best for the individual and what’s best for both collectively: the best joint outcome for the two prisoners is to remain silent and go to prison for only 2 years each, but no matter what the other prisoner does, a prisoner’s own prison sentence is lower if they confess and betray the other person. The Prisoner’s Dilemma can be generalised beyond the specific prison-context to any situation in which DC $>$ CC $>$ DD $>$ CD (the first letter indicates the first player’s choice and the second letter indicates the other player’s choice: DC = Defect when other Cooperates; CC = mutual Cooperation; DD = mutual Defection; CD = Cooperating when other defects); especially in iterated experiments, a second rule is implemented (2CC $>$ DC + CD) to ensure that taking turns defecting and cooperating isn’t the best long-term strategy. Another frequently studied 2 $\xd7$ 2 game is Chicken (Rapoport & Chammah, 1966; Smith & Price, 1973). Chicken (also known as Snowdrift and Hawk-Dove) is identical to the Prisoner’s Dilemma with one exception: DD and CD are swapped, such that for Chicken the payoffs are DC $>$ CC $>$ CD $>$ DD. This payoff swap changes the dynamics of the game: defection is no longer dominant. Chicken models a situation in which mutual destruction is worse than being taken advantage of and has been likened to nuclear warfare (Russell, 1959).

The Prisoner’s Dilemma might be the most famous 2 $\xd7$ 2 game, but it is only one of many. In a 2 $\xd7$ 2 game (Rapoport et al., 1976), 2 players each make a binary decision, leading to 4 possible outcomes. The different games can be defined by the order of their payoffs. Of all ordinal 2 $\xd7$ 2 games with strict preferences, 12 are symmetric (both players have the same payoffs; names from Bruns (2015); see Table 1).

Game . | 1 > . | 2 > . | 3 > . | 4 . |
---|---|---|---|---|

Chicken | DC | CC | CD | DD |

Battle | DC | CD | CC | DD |

Hero | DC | CD | DD | CC |

Compromise | DC | DD | CD | CC |

Deadlock | DC | DD | CC | CD |

Prisoner’s Dilemma | DC | CC | DD | CD |

Stag Hunt | CC | DC | DD | CD |

Assurance | CC | DD | DC | CD |

Coordination | CC | DD | CD | DC |

Peace | CC | CD | DD | DC |

Harmony | CC | CD | DC | DD |

Concord | CC | DC | CD | DD |

Game . | 1 > . | 2 > . | 3 > . | 4 . |
---|---|---|---|---|

Chicken | DC | CC | CD | DD |

Battle | DC | CD | CC | DD |

Hero | DC | CD | DD | CC |

Compromise | DC | DD | CD | CC |

Deadlock | DC | DD | CC | CD |

Prisoner’s Dilemma | DC | CC | DD | CD |

Stag Hunt | CC | DC | DD | CD |

Assurance | CC | DD | DC | CD |

Coordination | CC | DD | CD | DC |

Peace | CC | CD | DD | DC |

Harmony | CC | CD | DC | DD |

Concord | CC | DC | CD | DD |

The only difference between different ordinal 2 $\xd7$ 2 games is the rank of their payoffs and the resulting strategic decisions for the players (e.g., Nash equilibria). Consider the 4 games presented in matrix-form in Figure 2: the Prisoner’s Dilemma models conflict between what’s best for the individual (defect) and what’s best for both players combined (cooperate); for Chicken, getting exploited is not the worst option, mutual defection is, making defection the riskier option and cooperation safer; the reverse is true for Stag-Hunt, where cooperation can lead to the highest or the lowest payoff; Concord contains no real conflict: mutual cooperation is the best payoff and mutual defection the worst. Thus, 2 $\xd7$ 2 games collectively map various interdependent decisions and can be used for studying social interactions.

### 1.2. Matrix games and decomposed games

In most empirical studies, 2 $\xd7$ 2 games are displayed as the outcomes of the interdependent decision (Rapoport et al., 1976): if you choose C and the other person chooses D, you get 1 dollar and the other gets 7 dollars (see Figure 2a). This is usually displayed in a matrix and called the ‘matrix form’. But the outcomes can be decomposed into actions with consequences for self and other that are independent of what the other person chooses: if you choose C, you get 0 dollar and the other person gets 5 dollars, but if you choose D, you get 2 dollars and the other person gets 1 dollar (Figure 3 left). If both players decide between such decomposed options, the matrix form and the decomposed form describe the same 2 $\xd7$ 2 game, only changing the way the game is presented. For example, Figure 3 shows how if both players decide between the decomposed options just mentioned, the final outcomes are identical to the Prisoner’s Dilemma displayed in Figure 2.

The central difference between the matrix form and the decomposed form thus lies in their different emphasis on action and outcome. While the matrix form shows only the consequence of the players’ choices, the decomposed form shows only the value of the actions themselves, without explicitly stating the final outcomes. In matrix form, a decision has no intrinsic value and can only be evaluated in relation to the other’s decision; in decomposed form, each action carries an intrinsic value for oneself and the other. Decomposed games may thus be psychologically similar to allocation tasks: formally, they are 2 $\xd7$ 2 games, but they appear like a forced-choice allocation task, where each player chooses 1 of 2 possible allocations options for oneself and another person. The tasks of the social-value orientation literature are often described as ‘decomposed games’ (Kuhlman & Marshello, 1975; Murphy & Ackermann, 2013), but in these tasks usually only one person makes a decision, such that they are not games in the strict sense. In this article, we use the term ‘decomposed games’ to refer to a 2 $\xd7$ 2 game in which both players decide between 2 options. Decomposed games thus allow changing the way a game is framed, manipulating the story behind the game.

## 2. Decomposing different 2 × 2 games

### 2.1. Symmetric game, symmetric decomposition

This section answers the questions: 1) ‘which games can be decomposed?’ and 2) ‘if a game can be decomposed, what are possible decompositions?’.

From previous studies, we know that the Prisoner’s Dilemma can be decomposed. For example, Pruitt (1967) empirically tested the effect of different decompositions of the same Prisoner’s Dilemma on human cooperation rates. Pruitt decomposed the payoff matrix DC = 18, CC = 12, DD = 6, CD = 0 into several decomposed games, including the 2 displayed in Figure 4. These decompositions are psychologically quite different (in decomposition 1, you can either split $12 in half, or take $6 away from the other to get $12 yourself, but in decomposition 2 the decision lies between either giving $12 to the other player or taking $6 oneself), but if both players choose between the same decomposed options, the resulting game is the same. Pruitt found that participants had a 55% cooperation rate for the matrix-form and Decomposition 1 from Figure 4, and a 70% cooperation rate for Decomposition 2.

#### 2.1.1. Which symmetric 2 × 2 games can be decomposed symmetrically?

Pruitt found that for a Prisoner’s Dilemma to be decomposable, it has to fulfil the necessary condition CC - DC = CD - DD (which is equivalent to DD - DC = CD - CC and CC + DD = CD + DC). As we demonstrate in the Appendix, this necessary condition emerges as an algebraic consequence of the way decomposed games are set up and is independent of the order of payoffs: any 2 $\xd7$ 2 game can be decomposed symmetrically if and only if CC - DC = CD - DD.

What are the consequences of this necessary condition? First, we can use this rule to find out which games are decomposable in principle by taking CC - DC = CD - DD and searching for contradictions. As Table 2 shows, 4 games do not inherently contradict CC - DC = CD - DD: Deadlock, Prisoner’s Dilemma, Harmony, and Concord. These are the only strict ordinal symmetric 2 $\xd7$ 2 games that can be decomposed symmetrically. This logical explanation of why only 4 symmetric 2 $\xd7$ 2 games can be decomposed symmetrically can be complemented in a visually more intuitive way (Figure 5; for a related graphical representation of decomposed games, see Griesinger & Livingston, 1973): Fix one of the two decomposed options at an arbitrary point (the black dot in the centre) and let the other decomposed option vary freely. The resulting decomposed game depends only on the relative position of both options: for example, if the second point lands in the lowest right triangle (as depicted in blue) or highest left triangle, the resulting game is a Prisoner’s Dilemma; if it lands in the second lowest right triangle or the second highest left triangle, the game is Deadlock, and so on. If the freely-varying point were to land on the diagonal, horizontal, or vertical lines, the resulting game would not be a ordinal 2 $\xd7$ 2 games with strict preferences: e.g., if the freely-varying point were to land on the diagonal between Prisoner’s Dilemma and Deadlock, the resulting payoff matrix would be DC $>$ CC $=$ DD $>$ CD. In other words, the game would be between the Prisoner’s Dilemma and Deadlock.

Game . | 1 > . | 2 > . | 3 > . | 4 . | Potentially decomposable? . |
---|---|---|---|---|---|

Chicken | DC | CC | CD | DD | No: DC \(>\) CC \(\Rightarrow\) CC - DC \(<\) 0; CD \(>\) DD \(\Rightarrow\) CD - DD \(>\) 0 |

Battle | DC | CD | CC | DD | No: DC & CD \(>\) CC & DD \(\Rightarrow\) DC + CD \(\neq\) CC + DD |

Hero | DC | CD | DD | CC | No: same as for Battle |

Compromise | DC | DD | CD | CC | No: DC \(>\) DD \(\Rightarrow\) DD - DC \(<\) 0; CD \(>\) CC \(\Rightarrow\) CD - CC \(>\) 0 |

Deadlock | DC | DD | CC | CD | Yes |

Prisoner’s Dilemma | DC | CC | DD | CD | Yes |

Stag Hunt | CC | DC | DD | CD | No: CC \(>\) DC \(\Rightarrow\) CC - DC \(>\) 0; DD \(>\) CD \(\Rightarrow\) CD - DD \(<\) 0 |

Assurance | CC | DD | DC | CD | No: CC & DD \(>\) DC & CD \(\Rightarrow\) CC + DD \(\neq\) CD + DC |

Coordination | CC | DD | CD | DC | No: same as for Assurance |

Peace | CC | CD | DD | DC | No: DD \(>\) DC \(\Rightarrow\) DD - DC \(>\) 0; CC \(>\) CD \(\Rightarrow\) CD - CC \(<\) 0 |

Harmony | CC | CD | DC | DD | Yes |

Concord | CC | DC | CD | DD | Yes |

Game . | 1 > . | 2 > . | 3 > . | 4 . | Potentially decomposable? . |
---|---|---|---|---|---|

Chicken | DC | CC | CD | DD | No: DC \(>\) CC \(\Rightarrow\) CC - DC \(<\) 0; CD \(>\) DD \(\Rightarrow\) CD - DD \(>\) 0 |

Battle | DC | CD | CC | DD | No: DC & CD \(>\) CC & DD \(\Rightarrow\) DC + CD \(\neq\) CC + DD |

Hero | DC | CD | DD | CC | No: same as for Battle |

Compromise | DC | DD | CD | CC | No: DC \(>\) DD \(\Rightarrow\) DD - DC \(<\) 0; CD \(>\) CC \(\Rightarrow\) CD - CC \(>\) 0 |

Deadlock | DC | DD | CC | CD | Yes |

Prisoner’s Dilemma | DC | CC | DD | CD | Yes |

Stag Hunt | CC | DC | DD | CD | No: CC \(>\) DC \(\Rightarrow\) CC - DC \(>\) 0; DD \(>\) CD \(\Rightarrow\) CD - DD \(<\) 0 |

Assurance | CC | DD | DC | CD | No: CC & DD \(>\) DC & CD \(\Rightarrow\) CC + DD \(\neq\) CD + DC |

Coordination | CC | DD | CD | DC | No: same as for Assurance |

Peace | CC | CD | DD | DC | No: DD \(>\) DC \(\Rightarrow\) DD - DC \(>\) 0; CC \(>\) CD \(\Rightarrow\) CD - CC \(<\) 0 |

Harmony | CC | CD | DC | DD | Yes |

Concord | CC | DC | CD | DD | Yes |

The second consequence of the necessary condition is that any decomposable game requires a certain symmetry: using the first two formulations of the necessary condition (CC - DC = CD - DD and DD - DC = CD - CC), we can see that the difference between the first and second options has to be the same as the difference between the third and fourth options (see Figure 6). This places some limitations on which games can be decomposed: for example, if DC is increased, the game will only be decomposable if CC is increased equally, if DD is increased equally, or if CD is decreased equally.

The third consequence of the necessary condition is that any Prisoner’s Dilemma that is decomposable also abides by the second rule of the Prisoner’s Dilemma (2CC $>$ DC + CD). Using the third formulation of the necessary condition (CC + DD = CD + DC), it becomes clear that the first rule of the Prisoner’s Dilemma (DC $>$ **CC $>$ DD** $>$ CD) implies that CC $>$ 1/2(DC + CD), and therefore 2CC $>$ DC + CD. This means that the second rule of the Prisoner’s Dilemma holds for any decomposed decomposed Prisoner’s Dilemma, such that alternative cooperation and defection of both players cannot be the most beneficial strategy, even in an iterated setting.

#### 2.1.2. What decompositions are possible, and why are there infinitely many?

Pruitt also mentioned that if a Prisoner’s Dilemma payoff matrix is decomposable, then there are infinitely many possible decompositions. But how do the infinite decompositions relate to each other; can we choose freely which decomposition to use, or are these decompositions related in some systematic way? As above, the full explanation is in the Appendix, and the approach isn’t defined by the order of payoffs in the Prisoner’s Dilemma, so the other 3 decomposable games also have infinitely many decompositions once the necessary condition is fulfilled. From the algebraic formulation in the Appendix, we can create a table with the generic formula for symmetrically decomposing any symmetric 2 $\xd7$ 2 game (if it fulfills the necessary condition):

The decomposed options are defined by the payoffs of the game and $\gamma $, which can be chosen freely and is subtracted from payoffs for the self and added to payoffs for the other. Mathematically, the infinity of decomposed options per 2 $\xd7$ 2 game is trivial (x - x = 0), but from a practical perspective, this provides flexibility when designing experiments to alter the framing of the decomposed option.

As an example, Figure 8 displays different decompositions of the same payoff matrix. We use the Prisoner’s Dilemma with the payoff matrix DC = 7, CC = 5, DD = 3, CD = 1, and then use $\gamma $ values of (-2, -1, 0, 1, 2). The resulting decomposition lead to the same payoff matrix specified above (see Figure 8).

The specific value one selects for $\gamma $ will depend on multiple factors, such as the scientific question and the payoff matrix. To provide some guidelines, the total contributed points for C and D remain constant (because $\gamma $ is always added to the same extent that it is subtracted; in Figure 8 C always provides a total of 5 points and D always provides a total of 3 points), but $\gamma $ can affect other factors, such as absolute inequality for Self and Other allocations for the C and D options (in Figure 8 this is 1, 1, 3, 5, and 7 for C, and 5, 3, 1, and 1 for D), as well as advantageous and disadvantageous inequality (Fehr & Schmidt, 1999): in Figure 8, C switches from advantageous to disadvantageous inequality from example a to b, whereas D makes the same switch from example d to e; in example c, C and D have the same absolute inequality but C has disadvantageous inequality and D has advantageous inequality. The magnitude of $\gamma $ will likely be in a similar order of magnitude as the payoffs (i.e., if the payoffs are in the 100s, a $\gamma $ of 1 or 2 is unlikely to have much an effect on people’s decisions). Again, these questions will depend on the experimental question.

### 2.2. Asymmetric games and decompositions

So far, we have only dealt with symmetric games and symmetric decompositions: both players have the same options and the same potential outcomes. But real life doesn’t consist exclusively of symmetric situations. Outcomes can differ between people, the actions available to them can differ, or both. In the context of decomposed games, there can be asymmetric decomposed games and/or asymmetric decompositions. To account for decomposed games with asymmetries, either in payoff matrix or the decomposed options, we expand our previous section.

Conceptually, we can distinguish between 2 different kinds of asymmetries in the payoff structures of 2 $\xd7$ 2 games (asymmetric games): first, both players’ outcomes have the same payoff structure but with different values; second, the ordering of the payoffs differs between both players. Figure 9 displays both situations: first, in Figure 9a, the payoffs of both players are that of the Prisoner’s Dilemma, but the row player’s payoffs are multiplied by 10. Thus, for each player, the standard game-theoretic strategic considerations are the same as in a symmetric Prisoner’s Dilemma, but psychologically it might feel quite different. For example, if we consider different aspects of social-value orientation (Bogaert et al., 2008; Murphy & Ackermann, 2013), joint gain is now highest if the row player defects and the column player cooperates – but if a player cares most about reducing inequality, then the opposite would yield the best result (if the row player cooperates and the column player defects, the absolute difference between both players is only 3). Second, in Figure 9b, both players have the same values for their payoffs, but the payoffs of the column player are no longer in the order of the Prisoner’s Dilemma, but instead in the order of Concord. This combination of strict ordinal games constitutes a new game (Bruns, 2015). Thus, there are two different kinds of asymmetries of the payoff structure that we could incorporate into decomposed games.

Additionally, we would like to incorporate situations in which the decompositions (actions) differ between the players (asymmetric decompositions). We can use the same approach as before, but need to specify independent variables for each player, both for the payoffs and for the decomposition. We thus expand Figure 3 to Figure 10:

Figure 11 provides the summary of how to decompose any 2 $\xd7$ 2 game. As before, the full explanation is in the Appendix. The game can be symmetric or either type of asymmetric, and the decompositions can be symmetric or asymmetric. 2 parameters ($\alpha $ and $\beta $) can be chosen freely: as before, the self and other columns have to add up to a constant for the ultimate payoffs to be constant. The necessary condition CC - DC = CD - DD still holds, but separately for the payoff matrices of each player. Thus, asymmetric games are only decomposable if each of the players’ individual games is decomposable.

For asymmetric decompositions, the same principles apply for selecting the parameter as for symmetric decompositions, with the sole difference that now there are 2 separate parameters that can be chosen freely to accommodate a researcher’s methodological needs. In such asymmetric decompositions, selecting ideal parameters for $\alpha $ and $\beta $ is less intuitive than for symmetric decompositions because the Self and Other allocations for the two players no longer coincide; instead, each player’s final payoff receives its own parameter: $\alpha $ is for Player 2 (Self allocation for Player 2 and Other allocation for Player 1), and $\beta $ is for Player 1 (Self allocation for Player 1 and Other allocation for Player 2). As with symmetric decompositions, this flexibility allows researchers to vary different kinds of inequalities, but with the added flexibility that for asymmetric decompositions there can be further inequality between the two players’ total contributions.

As examples, Figure 12 shows a symmetric Prisoner’s Dilemma decomposed into asymmetric decompositions, where player 1 contributes more than player 2; Figure 13 shows decompositions for when the payoff structure differs between both players: player 1’s payoffs are of a Prisoner’s Dilemma, and player 2’s payoffs are of Concord.

### 2.3. Games with ties

So far, we only considered ordinal games with strict preferences, such that per player 2 payoffs cannot be the same. For example, for a game to be considered a Prisoner’s Dilemma, DC has to be larger than CC, the player cannot be indifferent about the order of the two outcomes. But just as we generalized symmetric games to asymmetric games, in real life different outcomes can be equally appealing. Thus, our final expansion includes games with ties between payoffs (e.g., DC $>$ CC $=$ DD $>$ CD).

As before, for a game to be decomposable, the necessary condition still holds for each players’ payoffs: DC + CD = CC + DD, even if outcomes are equal. Take a game with the payoff matrix (DC = 3, CC = 2, DD = 2, CD = 1). This game is between the Prisoner’s Dilemma (DC $>$ CC $>$ DD $>$ CD) and Deadlock (DC $>$ DD $>$ CC $>$ CD). Because DC + CD = CC + DD, the game is decomposable. Figure 14 shows two possible decomposed versions of this game, one symmetric and the other asymmetric.

Only games with ties that lie ‘between’ decomposable ordinal games can be decomposed (i.e., the dashed lines in Figure 5), including the special case where C = D and DC = CC = DD = CD (i.e., possible actions and outcomes are identical). Thus, decomposed games can not only incorporate symmetric and asymmetric games for ordinal games with strict preferences, but also for games with ties. This allows for even more flexible and nuanced experimental designs and thus expands the range of questions one can answer with decomposed games.

## 3. Conceptual differences between matrix games and decomposed games

When could one use the matrix-form and when could one use the decomposed form? Any specific response depends on the specific experimental question, but we can provide general guidelines that might aid deciding between these two ways of presenting 2 $\xd7$ 2 games in an experiment.

To our knowledge, not much research has systematically compared games in matrix form and decomposed form. Although some early studies compared the two (Evans & Crumbaugh, 1966; Messick & McClintock, 1968; Pruitt, 1967), these study focused on individual aspects: Evans and Crumbaugh compared participants’ cooperative choices in the matrix-form and one decomposed form and found that for that particular decomposition and payoffs, participants cooperated more in the decomposed than in the matrix form; Pruitt generalised symmetric decomposed games in the Prisoner’s Dilemma and found differences in cooperation rate between the matrix-form and decomposed form, and differences in cooperation rate between different decomposed forms of the same payoff matrix; Messick and McClintock found that decomposed games could be used to assess different motivational aspects of their participants. Those studies deal only with the symmetric Prisoner’s Dilemma (no other 2 $\xd7$ 2 games and no asymmetries between players) and mainly show that decomposed games can affect people’s decisions, without attempting to systematically study how or why decomposed games can affect people’s behaviour. Similarly, although the term ‘decomposed games’ has been used extensively in social-value orientation, these studies are not games but allocation decisions because only one person decides (Murphy & Ackermann, 2013) - and thus these studies also do not tell us anything about the differences between matrix form and decomposed form. Many of the differences we point out are thus ‘potential differences’, rather than ‘established differences’, and could be tested empirically in future studies.

First, the main conceptual difference between matrix form and decomposed form lies in their different foci: the matrix form emphasises outcomes, the decomposed form emphasises the actions. In matrix form, players decide between different outcomes, but in decomposed games players can base their decision on either the intrinsic value of an action itself (e.g., I prefer giving equally to both and will choose the action with the smallest difference between both players) or on the outcome (as in the matrix form). This distinction is related to the distinction between deontological ethics and consequentialism, where the former values the action itself higher and the latter values the outcome higher, independent of the action that went into the outcome (Alexander & Moore, 2023). This difference between action and outcome is also reminiscent of the distinction between procedural fairness and outcome fairness (Brockner & Wiesenfeld, 1996). Decomposed games might provide a new angle to study open questions in those fields.

Second, games might be more ecologically valid in decomposed form than in matrix form. In many situations in life, actions have a direct effect on oneself and someone else, independent of what the other person does. In the matrix form, however, actions only exist in the interdependent context; thus, the decomposed form might be more ecologically valid because they let people decide between actions with inherent value, which add up to a specific 2 $\xd7$ 2 game (Pruitt, 1967).

Third, from a practical perspective, games in decomposed form might be easier to understand for participants (and animals). Thus, for any experimental design that might benefit from a simpler task (e.g., if the participants/animals might struggle with the instructions; if there is little time in the experiment to explain the task; when testing children; if the rest of the experiment is already very complex), it might be useful to use the decomposed form. The simpler task structure might facilitate research that would otherwise not be possible. Studying cooperation and social decision-making in animals often requires a relatively simple experimental design (e.g., the car-driving task with rhesus macaques (Ong et al., 2020) or the rope-pulling task with elephants (Plotnik et al., 2011); using decomposed games may allow novel variants of these tasks. Caution is advised: any decomposition will affect the framing of the situation, which might have unintended side-effects.

## 4. Practical Guidelines

So far we have mainly considered theoretical aspects of decomposed games, such as what decomposed games are, which games can be decomposed, which decompositions are possible for any given payoff matrix. In this section, we summarise the main practical considerations for using decomposed games in experimental research (decomposed games can of course also be used for simulation studies, but our focus is empirical investigations), including their limitations.

One of the main considerations for decomposed games is the question of which values to choose for $\alpha $, $\beta $, and $\gamma $. We highlight two aspects that can be manipulated with these parameters: inequality, and valence.

*Inequality*. All three parameters can be used to change various aspects of inequality of the contributions (absolute, advantageous, disadvantageous) between the allocations for self and other. Thus, when selecting the ideal decompositions for a given payoff matrix, researchers ought to carefully consider whether changing these parameters might inadvertently have affected various aspects of inequality of the options, and whether this might affect people’s behaviour. Factors such as advantageous and disadvantageous inequality (Fehr & Schmidt, 1999) could easily be manipulated, such that decomposed games could be used to investigate to what extent the various aspects of inequality influence people’s decisions; unlike standard 2 $\xd7$ 2 games, decomposed games allow a disentangling of action and outcome, such that these aspects can be considered separately from each other. Additionally, for asymmetric decompositions, $\alpha $ and $\beta $ can be used to affect the total contributions for each player. For example, Figure 12 shows a symmetric game with asymmetric decompositions, such that both players receive the same potential outcomes, but Player 1 contributes many points and Player 2 total contributions are a net negative. Such asymmetries may affect people’s behaviour if they care not only about equality of outcomes but also equality of inputs.

*Valence*. The other factor, valence, can equally be affect by each of the three parameters: for many payoff matrices, changing $\alpha $, $\beta $, and $\gamma $ can lead to changing whether an allocation for self or other can lead to a categorical change between a gain or a loss. In an empirical study, we showed that losses and gains can have strong effects on people’s decisions to cooperate, leading to both increases and decreases of cooperation, depending on the context (Kuper-Smith & Korn, 2023). For those interested in how losses and gains affect cooperation, future studies could further investigate such questions in decomposed games: by changing the $\alpha $, $\beta $, and $\gamma $, we could shift the different decompositions relative to 0 without changing the valence of the outcomes. This could help ask further questions about how losses and gains affect social decisions, such as which types of losses matter most (loss of action or loss of outcome, for self or for other). Any researcher not interested in such questions ought to ensure that their change of $\alpha $, $\beta $, and $\gamma $ did not also lead to changes in behaviour due to changing the valence of the allocation options.

A further important question to consider is what information one wants to reveal to the participants (e.g., whether to show only the decomposition or also payoff matrix, whether the decompositions are symmetric or not, etc.). The question of how the visibility of decomposed options and resulting payoff matrices affects people’s cooperative choices was examined in a study (Pincus & Bixenstine, 1977) in which participants either saw the decomposed options alone or alongside the resulting payoff matrices. Showing the decomposed options alone or with the resulting payoff matrix was shown to alter people’s cooperation rates. Future studies could clarify why these effects occurred: which aspects matter, and how and why these aspects affect people’s cooperative choices. It could also be interesting to design studies in which players have different decomposed options, but are not made aware of this option until it is revealed, at which point one could see how people’s decisions and attitudes about the other person change.

Decomposed Games could also be used to study what aspects of a game people pay attention to, how salient different options are, and what features people use to make decisions. For example, one could present the payoff matrix alongside decomposed options of that payoff matrix and use eye-tracking (Polonio et al., 2015) to study which aspects people pay most attention to: do they attend most to the final outcomes, or to the actions that lead to them? Does this differ for different decompositions, and what factors influence this?

So far, we have only discussed decomposed game in the context of standard economic games. But decomposed games can also be embedded in an ecological context, which often have probabilistic outcomes over multiple steps. For example, previous research (Korn & Bach, 2015, 2018) has studied how people behave in foraging situations. So far, these studies are non-social studies of risky choices, but such foraging contexts could be expanded to include social decisions where two (or more) players make such decisions with outcomes for themselves and the other player. Any such game would then be a decomposed game, even with probabilistic outcomes. This could also help further the link between evolutionary, economic, and psychological game theory.

Any of the features mentioned so far, especially inequality and valence, could also be investigated in clinical populations. For example, patients with Borderline Personality Disorder show impaired social functioning (Jeung & Herpertz, 2014), and several studies found abnormal behaviour in the Ultimatum Game related to fairness (De Panfilis et al., 2019; Polgár et al., 2014; Thielmann et al., 2014), which could be investigated by altering the inequality of action and outcomes with decomposed games; valence has been linked to compulsivity, with some studies suggesting abnormalities in addiction (Mogg et al., 2003) and obsessive-compulsive behaviour (Sachdev & Malhi, 2005), which could be further investigated by manipulating the valence of the actions and outcomes. Thus, decomposed games could be used to further understand these conditions.

Decomposed Games are of course not without limitations. We highlight three main limitations, all of which are caused by the fixed way that decomposed games are necessarily set up. First, the biggest limitation of decomposed games is that not every game can be decomposed. Of the 12 symmetric ordinal 2 $\xd7$ 2 games with strict preferences, only 4 can be decomposed (Prisoner’s Dilemma, Deadlock, Harmony, and Concord); of the games with ties, only those ‘between’ the 4 ordinal games can be decomposed. This means that some of the most interesting 2 $\xd7$ 2 games, which are commonly used to study social dilemmas, cannot be decomposed, including Stag-Hunt and Chicken. This might also explain why decomposed games have almost exclusively focused on the Prisoner’s Dilemma so far: the other 3 decomposable games offer less of a social dilemma, in that, usually, one option is preferable to the other, which is less uniformly the case for the Prisoner’s Dilemma (or other games like Stag-Hunt or Chicken). Thus, given that only certain games can be decomposed, this limits the variety of games (and therefore social situations) that can be studied using decomposed games. Second, although decomposed games allow for lots of flexibility when choosing the right decomposed options for one’s experimental design, there are several factors that cannot be changed. For example, for symmetric decompositions, the total amount of points allocated to Self and Other for the C option is always equal to CC (because the CD from Self is cancelled out by the -CD from Other, as is the positive and negative $\gamma $ parameter; for D the total allocations is always equal to DC-CC+CD for the same reason). Thus, certain potentially interesting aspects one might like to vary are not possible to be varied independently with decomposed games (one could change CC, but this obviously also alters the final payoffs, rather than just the decomposed options). Thus, despite the general flexibility (especially for asymmetric decompositions), there are some limitations as to what can be altered. Third, decomposed games’ main strength, namely that it can change the framing of the game, is also a potential weakness: when setting up a decomposed game to study one factor, say how losses and gains affect people’s cooperative decisions (Kuper-Smith & Korn, 2023) in decomposed games, other factors can easily interfere, such as advantageous and disadvantageous inequality (Doppelhofer et al., 2021; Fehr & Schmidt, 1999) of the actions/decompositions: changing the parameters to alter one factor automatically will change other, often unintended, factors that could have large effects on people’s behaviour. Thus, whenever choosing a decomposition for one’s study, one has to be careful to consider what other factors are affected by one’s manipulation. While this is a generic problem of almost any experimental design, decomposed games are particularly likely to lead to unintended consequences, if not considered carefully.

## 5. Conclusion

Linearly-additive decomposed 2 $\xd7$ 2 games have existed for more than 50 years in the context of the Prisoner’s Dilemma but have been studied relatively rarely and, to our knowledge, have never been explored systematically. This is surprising, given the flexibility that they provide in assessing various aspects of social preferences, particularly in relation to inequality between two players in terms of actions and outcomes. In this article, we highlighted linearly-additive decomposed games by explaining their logic showing how they can apply beyond the Prisoner’s Dilemma, providing a way to calculate possible decompositions for a given payoff matrix (for symmetric and asymmetric games and decompositions), and providing some practical suggestions. Decomposed games allow for more flexible experimental designs, potentially enabling for ecologically more variable and realistic experimental set-ups.

## Author Contributions

BJKS wrote an initial short draft which both authors developed, expanded and edited.

## Funding

This work was supported by the Emmy Noether Research Group grant (392443797) from the German Research Foundation (DFG).

## Competing Interests

Both authors report no competing interests.

## Data Accessibility Statement

This is a theoretical article and contains no data.

## 6. Appendix

To keep the main article short, we present the relevant mathematics here. Any introductory course on Linear Algebra, such as the one by Gilbert Strang (https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/index.htm), should cover the following.

Why is CC - DC = CD - DD a necessary condition for a Prisoner’s Dilemma payoff matrix to be decomposable? In a decomposed game, the standard options of C and D indicate a specific amount of points each player gets (see Figure 1, right example). We can calculate the standard payoffs DC, CC, DD and CD the following way:

This can be written as:

The first column stands for $x1$, the second column for $x2$, and so forth. To simplify the notation, we write $A$ and $b$ as a single *augmented matrix*. We can now test whether this set of equations has a solution, and if so how many and under which conditions. We use Gaussian elimination to bring this matrix into the upper triangular form:

The last row now states that 0$x1$ + 0$x2$ + 0$x3$ + 0$x4$ = DD + CC - CD - DC. Therefore, this system of equations only has a solution (i.e., the matrix is decomposable) if:

which is equivalent to

Therefore, this set of equations does not have a solution if CC - DC $\u2260$ CD - DD. If this condition were not to be fulfilled, then the set of equations would run into internal contradictions. This is why Pruitt said that CC - DC = CD - DD is a necessary condition for a Prisoner’s Dilemma payoff matrix to be decomposable.

To find out what all possible decompositions for a given payoff matrix is, we need to solve the set of equations. We take the upper triangular matrix and bring it to the row-reduced echelon form. Given that we are only interested in games that are actually decomposable, we know that the necessary condition of CC - DC = CD - DD is fulfilled. To simplify the set of equations, we can substitute DD + CC - CD - DC with 0:

We now bring this form into the reduced row echelon form by using Gauss-Jordan elimination:

By finding the particular and the special solution, we can now find the generic solution to this set of equations:

To make it easier to use, we reformatted this equation in Table 2. This solution also explains why if there is one decomposition, there are infinitely many decompositions: we can take a particular decomposition, and then shift the decomposition such that $x1$ and $x3$ are decreased by $\gamma $ to the same degree that $x2$ and $x4$ are increased. In other words, if for both options the self-payoff is decreased as much as the other-payoff is increased, the resulting payoff matrix remains constant.

At no point did we specify the order of the payoffs. This means that the results from this section hold true independent of which game is being used. As we show in the main text, only some games can abide by the necessary condition, but for those games, the solutions offered here apply equally.

So far we have assumed that both players play the same game with the same decomposed options. Using the same approach as for symmetric games, we can now solve the system without those assumptions and thus incorporate asymmetric games and asymmetric decompositions. We skipped the steps here, but the approach is the same as above, just with more variables:

As before, we now get the necessary condition that DC + CD - CC - DD = 0, but for both players separately. This means that both players can have different payoff matrices, but they still have to each abide by the necessary condition for a 2 $\xd7$ 2 game to be decomposable.

Again, to get a generic formula for calculating the decompositions, we solve this set of equations:

As before, we reformatted this into a more usable format in Figure 11.

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