The PD game and SERS-driven choices

The PD game is a well-known model of cooperation and confrontation (Flood, Reference Flood1958; Rapoport and Chammah, Reference Rapoport and Chammah1965; Axelrod, Reference Axelrod1984). It is described by a two-by-two payoff matrix where each of two players may choose either a cooperative or a confrontational alternative (Fig. 1(a)). If both parties choose to cooperate, each player obtains the Reward (R) payoff; but if both choose to confront, each player obtains the Punishment (P) payoff. However, if one of the players chooses to confront while the other chooses to cooperate, the former obtains the Temptation (T) payoff and the latter obtains the Sucker's (S) payoff, where T > R > P > S. As derived from these inequalities, confrontation is a dominant strategy that allows players to protect themselves from exploitation (because P > S) and still retain the option to exploit a trusting opponent (because T > R). Many other decision rules, such as MaxiMin, MaxiMax (Rapoport, Reference Rapoport1967) or the joint perspective expressed by the Nash equilibrium (Nash, Reference Nash1950; Luce and Raiffa, Reference Luce and Raiffa1957) concur with this conclusion, recommending PD players to always confront their opponents. In contrast to these principles, behavioural choices made by human participants show that while some players choose to confront, many others choose to cooperate (Rapoport and Chammah, Reference Rapoport and Chammah1965; Sally, Reference Sally1995; Bornstein, Reference Bornstein2003, Reference Bornstein, Suleiman, Fischer, Budescu and Messick2004), suggesting that human behaviour is driven by other principles.

Fig. 1. Panel a: the Prisoner's Dilemma game. Left and right payoffs in each cell indicate the respective payoffs for the row and column players. The game is defined by the inequalities: T > R > P > S (and in some experiments also requires 2R > S + T; Flood, Reference Flood1958; Rapoport and Chammah, Reference Rapoport and Chammah1965). Panel b: a generic matrix structure showing two alternatives for the row and column players and their corresponding payoff values – V(row and column choices).

Bringing together theoretical rationality and observed behaviours, the SERS theory considers both the payoff values of the game and the extent of subjectively perceived strategic similarity with the opponent – the predicted prospects of both parties choosing identical alternatives in the forthcoming interaction, as subjectively experienced by each individual (Fischer, Reference Fischer2009, Reference Fischer2012; Fischer et al., Reference Fischer, Levin, Rubenstein, Avrashi, Givon and Oz2022; Fischer and Savranevski, Reference Fischer and Savranevski2023). Using subjective estimates of strategic similarity, denoted by p _s, and its complementary value, 1 − p _s (indicating the prospects of the opponent making dissimilar choices), players may, explicitly or rather implicitly, compute the expected values (EVs) for both possible choices of cooperation and confrontation. The EV of cooperation is given by EV_cooperation = R × p _s + S × (1 − p _s), and the EV of confrontation is given by EV_{confrontation} = P × p _s + T × (1 − p _s). Comparing EV_cooperation with EV_{confrontation} allows choosing the alternative that provides a higher EV. The derived decision rule may be expressed as: cooperate if R × p _s + S × (1 − p _s) > P × p _s + T × (1 − p _s), and confront if R × p _s + S × (1 − p _s) < P × p _s + T × (1 − p _s).

Separating the payoffs of the interaction (i.e. T, R, S, P) from opponent's strategic similarity perception, p _s, SERS's decision rule may be rewritten as: cooperate if p _s > (T − S)/(T − S + R − P), and confront the opponent if p _s < (T − S)/(T − S + R − P). Further defining the critical similarity threshold, p _s*, which marks the switching point between cooperation and confrontation by $p_{\rm s}\ast = ( T-S) /( T-S + R-P)$, we obtain an abridged decision rule that recommends cooperation whenever $p_{\rm s} > p_{\rm s}\ast$, confrontation whenever $p_{\rm s} < p_{\rm s}\ast$ and being indifferent if $p_{\rm s} = p_{\rm s}\ast$.

SERS's solution has been shown to correctly predict human choices in critical single-shot games, including PD, Chicken and Battle of the Sexes (Fischer, Reference Fischer2009, Reference Fischer2012; Fischer and Savranevski, Reference Fischer and Savranevski2023). It has also been developed into an elaborate algorithm that outperforms many powerful algorithms in evolutionary computer simulations (Fischer et al., Reference Fischer, Frid, Goerg, Levin, Rubenstein and Selten2013).

Extending SERS to account for all two-by-two games

Examining SERS across all two-by-two games provides a comprehensive description of SERS's strategic recommendations. More specifically, it allows distinguishing between three types of games: (i) similarity sensitive games, for which the optimal choice depends, not only on the payoffs, but also on the perceived level of similarity with the opponent, p _s, (as shown for the PD game). In other words, one of the alternatives should be preferred if p _s < p _s*, while the other alternative should be preferred if p _s > p _s*. The level of p _s* is dependent only on the payoffs of each specific game (and may therefore differ between games). The level of perceived strategic similarity, p _s, is subjectively estimated by each player, expressing the prospects of the opponent making identical choices (and may therefore differ between players). Importantly for the hereby considered humans–extraterrestrials interaction, a player expecting to play a similarity sensitive game may attempt influencing the choice of the opponent by messaging indications of strategic similarity that motivate opponent's cooperation. (ii) Games that are non-similarity-sensitive where SERS's decision rule recommends choosing the same alternative regardless of the estimated level of p _s. (iii) Games that are similarity sensitive for one of the players (either the row or the column player) but not for the other.

To quickly identify whether games are similarity sensitive (independently for each player), one may compare the preferred choice made under the assumption of p _s = 1 (i.e. both players are assured to choose identical alternatives) and the preferred choice made under the assumption of p _s = 0 (i.e. both players are assured to choose different alternatives). If the preferred choices differ, the game is similarity sensitive from the perspective of the player whose payoffs are considered, and vice versa.

SERS computation requires identifying the cells that constitute players' similar alternatives. Therefore, it is helpful to standardize all games by arranging the cells of each matrix in a fixed order that assures similar and dissimilar alternatives are located in designated respective cells, allowing to correctly associate similar-alternative cells with p _s, and dissimilar-alternative cells with 1 − p _s. Notice that switching rows, columns or both rows and columns does not change the game but only its presentation. Games that are not fully symmetric do not offer identical payoffs to both players. For these games, the similarity of the alternatives is not reflected by identical payoffs of each player, but by the relative proximity of the payoffs. In other words, by choosing similar alternatives in asymmetric games, players obtain payoffs that are not identical, but are more similar to each other than the payoffs obtained by choosing dissimilar alternatives.

Figure 1(b) shows a generic two-by-two game structure, with row and column players' alternatives denoted by A and B, where AA and BB represent similar alternatives. The payoff values for row and column players are denoted by V _row, and V _column, respectively. Row player's payoffs are defined as: V(AA)_row, V(AB)_row, V(BA)_row, V(BB)_row; and column's player payoffs are defined as: V(AA)_column, V(AB)_column, V(BA)_column, V(BB)_column.

After correctly aligning the cells in each matrix, we rewrite SERS's decision rule (for the row player) as:

Choose A if p _s × V(AA) + (1 − p _s) × V(AB) > p _s × V(BB) + (1 − p _s) × V(BA), choose B if p _s × V(AA) + (1 − p _s) × V(AB) < p _s × V(BB) + (1 − p _s) × V(BA), otherwise be indifferent. Further defining p _s* = [V(BA) − V(AB)]/[V(BA) − V(AB) + V(AA) − V(BB)] generates the abridged decision rule: choose A if p _s > p _s* and choose B if p _s < p _s* (Fig. 1(b)). Note that deriving the parallel decision rule for the column player requires switching V(AB) and V(BA).

Notably, players who expect their opponent to play a similarity sensitive game may try to influence the way they are being perceived by the opponent. They may attempt to convey a higher extent of strategic similarity, consequently expecting the opponent to choose a more cooperative alternative. If opponent's personal and social properties (e.g. beliefs, goals, traits, history or culture) are known, one may emphasize those properties that are more similar to one's own properties, and use them as indicators or proxies of strategic similarity. If no information is available, as in the case of unknown extraterrestrial entities, strategic similarity signals may still be conveyed by relying on rather fundamental concepts and universal properties (see Discussion).

Figure 2 illustrates three games played by two parties, denoted as humans and extraterrestrials, labelled in accord with the property of the game being similarity sensitive for both, one or none of the parties. Note that choosing an SERS-based alternative does not require calculating the games' similarity threshold or determining whether the game is similarity sensitive. In practice, each player only needs to correctly associate the probability of strategic similarity, p _s, with the corresponding payoffs (and the complimentary probability 1 − p _s with its corresponding payoffs) and compute the EV for each of the two alternatives. Finally, the player should compare both EVs and choose the alternative that provides the higher one. As shown in previous studies, actual human choices are highly correlated with this computational model, meaning that human individuals naturally and intuitively (without engaging in formal computations) associate similarity perceptions with the corresponding payoffs (Fischer, Reference Fischer2009, Reference Fischer2012; Fischer and Savranevski, Reference Fischer and Savranevski2023).

Fig. 2. Examples of two-by-two game matrices, here described as games taking place between humans and extraterrestrials. Humans choose between alternatives A and B, and extraterrestrials choose between alternatives α and β. Each cell shows the payoff values obtained by humans (left) and by extraterrestrials (right). EV(A) and EV(B) denote the expected value for the human party when choosing alternatives A or B, and p _s denotes the probability the human party assigns to the prospects of both parties making identical choices (i.e. either Aα or Bβ). Matrix a shows a game that is similarity-sensitive for both parties. Matrix b shows a non-similarity sensitive game. Matrix c shows a game that is similarity-sensitive only for the human party.

Simulations

To simulate a representative and comprehensive set of possible two-by-two interactions, we randomly sample payoffs from a quasi-continuous and equiprobable range (0, 1, …, 99, 100) to generate 10⁵ games, representing an illustrative and sufficiently large sample of games. We then distinguish between strategic similarity expectations of the parties, each assuming 11 levels (p _s = 0, 0.1, …, 0.9, 1), and compute SERS-based EVs of the human party for all 10⁵ games × 11 levels of human similarity expectations × 11 levels of extraterrestrial similarity expectations. Before computing the SERS-based EVs (Fig. 2), the simulation software examines each game and properly aligns the alternatives of both parties by determining which human alternative best corresponds to which extraterrestrial alternative, allowing to correctly associate p _s and 1 − p _s probabilities with similar and dissimilar choices. The alignment may be tested by applying a correlation-based or a difference-based procedure. Both methods compare the two possible configurations of the alternatives and test the extent to which the players face the same strategic problem under each of both possibilities, thus pointing to the more similar and to the more dissimilar choices. Notably, both configurations represent the same game; therefore the realignment does not change the strategic nature of the encounter (see Supplementary materials). Since both the correlation-based and the difference-based alignment procedures generate rather identical outcomes, the hereby reported results are derived from the correlation-based alignment method (the parallel results computed using the difference-based alignment are presented in the Supplementary materials). After alignment, the software computes the EVs for each alternative and selects the alternative with the higher EV, separately for humans and for extraterrestrials, thus determining both parties' preferred choices. Finally, the actual payoffs in the chosen cells (determined by the intersection of the selected alternatives) are averaged for all different combinations of human and extraterrestrial strategic similarity perceptions. The entire process is illustrated in Fig. 3.

Fig. 3. Illustration of the simulation process. Random payoff values are sampled to generate 10⁵ matrices. Each matrix is then separately tested. If necessary, an alignment protocol is activated according to the correlation-based method (see also the difference-based method in the Supplementary materials). Each matrix is associated with all tested p _s levels (11 strategic similarity perceptions for humans, and 11 perceptions for extraterrestrials), allowing calculating SERS-based EV(A), EV(B), EV(α) and EV(β). The alternatives with the higher EVs are separately selected for humans and for extraterrestrials, allowing to determine the jointly chosen cell and to assign its payoffs to the respective parties.

Figure 4 depicts simulation results showing the expected human payoffs across 11 possible human-held similarity perceptions (p _s = 0, 0.1, 0.2, …, 0.9, 1), each averaged across 11 possible levels of extraterrestrial similarity perceptions (p _s = 0, 0.1, 0.2, …, 0.9, 1), for a total of 1.21 × 10⁷ games. Panels a and b present separate results for similarity sensitive games, non-similarity-sensitive games, human-only similarity-sensitive games and extraterrestrial-only similarity-sensitive games (each game category comprises approximately 25% of the simulated sample); both panels also show the average payoffs across all game types. Panel a shows that the maximal human mean payoff is obtained when making decisions under the assumption of p _s = 0.5. Panel b shows that the maximal human mean payoff is obtained when extraterrestrials expect a maximal strategic similarity with humans (p _s = 1). Panel c depicts a two-dimensional heat map that shows human mean payoffs for all tested combinations of human and extraterrestrial similarity expectations, revealing a clear strategic sweet spot at the {0.5, 1} coordinate, thus indicating that the maximal mean human payoff is obtained when humans make decisions while assuming 50% strategic similarity with extraterrestrials and extraterrestrials make decisions while assuming 100% strategic similarity with humans. Moreover, the {0.5, 1} coordinate provides not only an optimal average expected payoff, but also the best possible payoff for 73% of all simulated games (i.e. no other coordinates provide better payoffs), as well as the lowest variance of human expected payoffs across all games (SD = 24.6; see simulation results in the Supplementary materials). These results suggest that as long as no strategic similarity level with extraterrestrials can be specified, risk-neutral payoff-maximizing humans should: (i) make decisions while assuming p _s = 0.5, and (ii) direct all messaging attempts to convey a strategic similarity level of p _s = 1.

Fig. 4. Simulation results. Panel a depicts mean human payoffs across various human expectations of extraterrestrial strategic similarity levels, calculated separately for each game type and across all games. Panel b depicts mean human payoffs across various extraterrestrial expectations of human strategic similarity levels, calculated separately for each game type and across all games. Panel c depicts a two-dimensional heat map of human mean payoffs calculated across 11 human × 11 extraterrestrial probabilities of similarity. Low and high human payoffs are denoted by shades of red and blue, respectively.