Fig 1.
Naturalistic example of a transparent two-player game: Group foraging in monkeys.
Two monkeys are reaching for food in two locations that are at some distance so that each monkey can take only one portion. At one location are grapes (preferred food), at the other—a carrot (non-preferred food). (A) Initially both monkeys move toward grapes. (B) Monkey 1 observes Monkey 2 actions and decides to go for the carrot to avoid a potential fight. (C) Next time Monkey 1 moves faster towards the grapes, so Monkey 2 swerves towards the carrot. Coordinated behaviour in such situations has the benefit of higher efficiency and avoids conflicts. This example shows that transparent game is a versatile framework that can be used for describing decision making in social contexts.
Fig 2.
Payoff matrices for Prisoner’s Dilemma and (Anti-)Coordination Game.
(A) In Prisoner’s Dilemma, players adopt roles of prisoners suspected of committing a crime and kept in isolated rooms. Due to lack of evidence, prosecutors offer each prisoner an option to minimize the punishment by making a confession. A prisoner can select one of the two actions (A1 or A2): either betray the other by defecting (D), or cooperate (C) with the partner by remaining silent. The maximal charge is five years in prison, and the payoff matrix represents the number of years deducted from it (for instance, if both players cooperate (CC, upper left), each gets a two-year sentence, because three years of prison time have been deducted). The letters R,T,S and P denote payoff values and stand for Reward, Temptation, Saint and Punishment, respectively. (B) In the (Anti-)Coordination Game variant known as Bach-or-Stravinsky and as Hero [27–29]) two people are choosing between Bach and Stravinsky music concerts. Player 1 prefers Bach, Player 2—Stravinsky, hence, there is an inherent conflict about which concert to choose; yet, above all both prefer going to the concert together. Thus the aim of the players is to coordinate (either on Bach or on Stravinsky), which assures maximal joint reward for the players. Players can either insist (I) on their own preference or accommodate (A) the preference of the partner. In these terms, the outcome coordination (attending the same concert) is achieved by selecting complementary actions: either (I, A) or (A, I), which justifies the name: “anti-”coordination. For example, when both agents coordinate on Bach, Player 1 insists, while Player 2 accommodates (I, A). In the “Methods”, we consider also a more general class of (anti-)coordination games, encompassing Hawk-Dove (or Chicken) and Leader.
Fig 3.
Frequency of establishing effective cooperation and the forfeit reward in the iterated Prisoner’s Dilemma (iPD) and in the iterated (Anti-)Coordination Game (i(A)CG).
We performed 80 runs of evolutionary simulations tracing 109 generations of iPD and i(A)CG players. Agents with successful strategies reproduced themselves (had higher fraction in the next generation), while agents with unsuccessful strategies died out, see “Methods” for details. We considered a run as “cooperative” if the average payoff across the population was more than 0.9 times the pay-off of 3 units for cooperative behaviour in iPD [24], and more than 0.95 times the pay-off of 3.5 units for cooperative behaviour in the i(A)CG (i.e., 90% and 95% of the maximally achievable pay-off on average over both players). For i(A)CG we set a higher threshold due to the less competitive nature of this game. (A) In iPD cooperation was quickly established for low probability to see the partner’s choice psee, but it took longer to develop for moderate psee and it drastically decreased for high psee. (B) In contrast, for i(A)CG frequent cooperation emerges only for high visibility. The small drop in cooperation at psee = 0.4 is caused by a transition between two coordination strategies (see main text). (C) The forfeit payoff (maximal possible average payoff of the population minus actual average payoff obtained by the population) further illustrates the same tendencies: higher transparency reduces effectiveness of cooperation in iPD but increases in i(A)CG.
Table 1.
Relative frequencies of strategies that survived for more than 1000 generations in the iterated Prisoner’s Dilemma.
The frequencies were computed over 109 generations in 80 runs. The frequency of the most successful strategy for each psee value is shown in bold.
Fig 4.
iPD strategies in the final population.
Strategies are taken for the 109-th generation and averaged over 80 runs. This figure characterizes the final population as a whole and complements Table 1 representing specific strategies. (A) Strategy entries s1, …, s4 are close to (1001) for psee = 0.1, …, 0.3 demonstrating the dominance of WSLS. Deviations from this pattern for psee = 0.0 and psee = 0.4 indicate the presence of the GTFT (1a1b) and FbF (101b) strategies, respectively. For psee ≥ 0.4 strategy entries s1, …, s4 are quite low due to the extinction of cooperative strategies. (B) Entries s5, …, s8 are irrelevant for psee = 0.0 (resulting in random values around 0.5) and indicate the same WSLS-like pattern for psee = 0.1, …, 0.3. Note that s6, s7 > 0 indicate that in transparent settings WSLS-players tend to cooperate seeing that the partner is cooperating even when this is against the WSLS principle. The decrease of reciprocal cooperation for psee ≥ 0.4 indicates the decline of WSLS and cooperative strategies in general. (C) Entries s9, …, s12 are irrelevant for psee = 0.0 (resulting in random values around 0.5) and are quite low for psee = 0.1, …, 0.3 (s12 is irrelevant in a cooperative population). Increase of s9, …, s11 for psee ≥ 0.4 indicates that mutual cooperation in the population is replaced by unilateral defection.
Fig 5.
Analytical pairwise comparison of iPD strategies.
For each pair of strategies the maps show if the first of the two strategies increases in frequency (up-arrow), or decreases (down-arrow) depending on visibility of the other player’s action and the already existing fraction of the respective strategy. The red lines mark the invasion thresholds, i.e. the minimal fraction of the first strategy necessary for taking over the population against the competitor second strategy. A solid-line invasion threshold shows the stable equilibrium fraction which allows coexistence of both strategies (see “Methods”). Dashed-line invasion thresholds indicate dividing lines above which only the first, below only the second strategy will survive. (A) WSLS has an advantage over GTFT
: the former takes over the whole population even if its initial fraction is as low as 0.25. (B) GTFT coexists with (prudent) version of cooperative strategy AllC (1111; 1111; 0000), which is more successful for psee ≥ 0.1. (C,D) L-F
performs almost as good as GTFT and WSLS, (E) but can resist the AllD strategy (0000; 0000; 0000) only for high transparency. (F) Note that WSLS may lapse into its treacherous version,
. This strategy dominates WSLS for psee > 0 but is generally weak and cannot invade when other strategies are present in the population. Notably, when treacherous WSLS takes a part of the population, it is quickly replaced by L-F, which partially explains L-F success for high psee.
Fig 6.
Frequencies of strategies that survived for more than 1000 generations after they emerged in the iterated Prisoner’s Dilemma population as function of reward R for mutual cooperation.
Data exemplified for psee = 0.3 and for psee = 0.5. Values of T, S and P are the same as in Fig 2, values of R are in range (S + T)/2 < R < T that defines the Prisoner’s Dilemma payoff. The frequencies were computed over 109 generations in 40 runs. We describe as “other cooperative” all strategies having a pattern (1*1*;1***;****) or (1**1;1***;****) but different from WSLS, TFT and FbF. While for psee = 0.3 population for low R mainly consists of defectors, for psee = 0.5 L-F provides an alternative to defection. For R ≥ 3.2 mutual cooperation becomes much more beneficial, which allows cooperative strategies to prevail for all transparency levels. Yet higher transparency reduces cooperation for all values of R. Note that the higher R is the less specific the cooperative strategies are. Indeed, for high R cooperation is much more effective than other types of behaviour, which makes all cooperative strategies (including even unconditional cooperation) evolutionary successful (we refer to [24] for a similar result in the case of sequential iPD).
Fig 7.
i(A)CG strategies in the final population.
Strategies are taken for the 109-th generation and averaged over 80 runs. This figure characterizes the final population as a whole and complements Table 2 representing specific strategies. (A): Strategy entries s1, …, s4. The decrease of the s2/s3 ratio reflects the transition of the dominant strategy from challenging to turn-taking for psee = 0.0, …, 0.4. For psee = 0.5 the dominance of the Leader-Follower strategy is indicated by s2 = s3 = 1. (B) Entries s5, …, s8 are irrelevant for psee = 0. Values of s6, s7 decrease as psee increases, indicating an enhancement of cooperation in i(A)CG for higher transparency (s8 is almost irrelevant since mutual accommodation is a very rate event, and s5 is irrelevant for a population of L-F players taking place for psee = 0.5). (C) Entries s9, …, s12 are irrelevant for psee = 0. The decrease of the s10/s11 ratio for psee = 0.1, …, 0.4 reflects the transition of the dominant strategy from challenging to turn-taking.
Table 2.
Relative frequencies of strategies that survived for more than 1000 generations in the (Anti-)Coordination Game.
The frequencies were computed over 109 generations in 80 runs. The frequency of the most successful strategy for each psee value is shown in bold.
Fig 8.
Analytical pairwise comparison of i(A)CG strategies.
For each pair of strategies the maps show if the first of the two strategies increases in frequency (up-arrow), or decreases (down-arrow) depending on visibility of the other player’s action and the already existing fraction of the respective strategy. The red lines mark the invasion thresholds, i.e. the minimal fraction of the first strategy necessary for taking over the population against the competitor second strategy. Solid-line invasion thresholds show the stable equilibrium fraction which allows coexistence of both strategies (see “Methods”). Dashed-line invasion thresholds indicate dividing lines above which only the first, below only the second strategy will survive. In all strategies, 1 stands for 0.999 and 0—for 0.001, the entries s9 = … = s12 = 1 are the same for all strategies and are omitted. (A) Turn-taker (q01q; 0000) with q = 5/8 for psee > 0 outperforms Aggressive Challenger , (B) but not Challenger
. (C) Challenger can coexist with Aggressive Challenger for low transparency, but is dominated for psee > 1/3. (D) Leader-Follower (1111; 0000) clearly outperforms Turn-taker for psee > 0.4 and (E,F) other strategies for psee > 1/3.
Fig 9.
Evolutionary dynamics of iPD-population consisting of two types of players: With WSLS and AllD strategies.
(A) Initially, both types have the same frequency, but after 40 generations the fraction of WSLS-players x1(t) converges to 0 for probabilities to see partner’s choice psee = 0.0, 0.2 and to 1 for psee = 0.4, 0.5. (B) This is due to the decrease of the invasion threshold h1 for WSLS: while h1 = 1 for psee = 0 (AllD dominates WSLS and the fraction of WSLS-players unconditionally decreases), AllD and WSLS are bistable for psee > 0 and WSLS wins whenever x1(t)>h1. Arrows indicate whether frequency x1(t) of WSLS increases or decreases. Interestingly, h1 = 0.5 holds for psee ≈ 1/3, which corresponds to the maximal uncertainty since the three cases (“Player 1 knows the choice of Player 2 before making its own choice”; “Player 2 knows the choice of Player 1 before making its own choice”; “Neither of players knows the choice of the partner”) have equal probabilities.