1 Introduction
Just over half a century ago, J. von Neumann and O. Morgenstern Neumann1944 probably had no idea of the extent that their work would have, not only on applications in the economic behaviour but also in several areas of biology, ecology, and mainly in the evolutionary theory of the species Maynard Smith 1982 culminating in the replicator dynamics Hofbauer2003 ; Nowak2006 and its branches (see, for example, the Refs. Hauert2005 ; Melbinger2010 ). In fact, the game theory still persists as an important theory of mathematical models which studies the choices of optimal decisions under conflict or combat conditions. However, long before the evolutionary aspects were studied in this context, the economical applications were the primordial motivations of this theory, and leveraged the construction of suitable concepts as Nash equilibrium, that contemplates the existence of an equilibrium of mixed strategies in noncooperative games Nash . The idea was very simple but brilliant: in such equilibrium condition, when there are two or more players, no player will gain by unilaterally changing its strategy. J. Nash was beyond and extended the concept to cooperative games by performing a reduction to noncooperative games. His important works culminated in the Nobel Memorial Prize in Economic Sciences jointly to R. Selten and J. Harsany Harsanyi and Selten .
An important concept in game theory arises when the game can be repeated numerous times. This game is sometimes referred in literature as dynamic game, or simply repeated game, in which the iteration effects can be understood under two different theories: the deterministic and stochastic game theories. One more interesting kind of dynamic game theory is one that covers the stochastic case, and was introduced and formalized by Shapley in 1953 Shapley1953 ; Mertens1981 ; Solan2015 .
Looking in the other direction, games where the profit depends on the number of investors have a direct similarity with the reality in the business world. Here, the paradigmatic El farol bar problem (EFBP), which is a simple model that shows how (selfish) players cooperate with each other in the absence of communication, arises as an important starting point Arthur1994 . In that problem, a set of people must decide if they go to a bar or stay at home. If they decide to go out and the bar is too crowded, they probably will not have a funny night and stay at home would be more interesting. They will have (or not have) a funny night according to a threshold on the number of people in the bar. If less than a fraction (a threshold) of the population decide to go to the bar, they will have a better time than ones which decided to remain in their houses. On the other hand, if more than a fraction of the population go to the bar, they will have bad times when compared with the comfort of their homes.
Two physicists saw the potential of this idea, and a formalization of the EFBP was given by Challet and Zhang Challet1997
in 1997, the socalled Minority Game (MG) which was proposed as a rough model to describe the price fluctuation of stock markets. In that game, an odd number of players have to choose one of two options independently at each turn. The players who end up on the minority side win the game. Following this line, another multiple player game (strongly based on experimental economics of the public goods), is the public goods game (PGG). In that game, multiple players can contribute to a common fund with or without the presence of communication in order to obtain benefits, and all money collected is doubled, or triplicate, or simply corrected according to some multiplicative factor. This game has been studied in many contexts by including optional participation obtaining a fixed income by opting not to invest in the public good, spatial diffusion effects and many other ingredients, both when studying many public goods games
Szabo2002 ; Hauert2003 ; Hauertetal2002 ; Pablo2017 and when considering only one public good (see, for example, SilvaIJBC2010 ; SilvaBJP2008 ; SilvaLNE2006 ; SilvaPhysA2006 ). Differently from MG where the payoffs of players are inversely proportional to the number of persons which choose the same group, in PGG the payoff depends on how many persons invest in the fund.The important feature of the MG is the absence of communication and noise. However, in general, real situations have properties such as misinformation, noise, selfishness, altruism, and some of these ingredients can coexist in more complex scenarios. In this work, we propose a new game that captures some aspects of both PGG and MG by taking into account some of those properties as well as multiplayer games in order to show how the altruism can affect the information of a group and change their payoffs. However, we consider very different constraints, with the payoffs being determined probabilistically and according to the information (and its propagation) controlled only by some players. We call it the boxers game, since it was partly based on the idea of the payoffs obtained by gamblers in boxing matches. However, the motivations can be easily extended to include, for instance, horserace or even gambling on the stock exchange, or even more general situations.
The paper is organized as follows: In the next section, we formulate the model. In Sec. 3, we present our main results which are separated in two parts. The first one considers the static, or fixed, investments and the sampling of payoffs are analysed through Monte Carlo (MC) simulations and analytical results. The second part of our results is related to the dynamic investments in which a simple evolutionary aspect of the game, based on reaction of the players according to their profit, is studied via MC simulations. Finally, our summaries and conclusions are presented in Sec. 4.
2 The model
In this approach, we consider a person game in a scenario where each player must choose between two boxers (groups): or . Some players (with density ) know that the boxer defeats the boxer with probability (denoted as ) given, for example, their skills. Therefore, boxer defeats boxer (denoted as ) with probability such that is the draw probability. Here, it is important to mention that the information is incomplete since is not exactly equal to 1 (in this case should be necessarily equal to ). The misinformation, which has an important role in the payoff of the players, was firstly formalized by the Nobel prize laureate in Economic Sciences, John C. Harsanyi Harsanyi . However, we propose a different approach: by considering the similarities with a boxing match, we establish that the payoff of the player in a group or depends on some features considering some protocols that remember both PGG and MG. So, we set up three important fundamentals that govern our game:

MG protocol: in the case of victory, the payoff of the players (winners) depends on how many persons have bet in the losing group;

PGG protocol: the profit obtained for the group must be divided equally among the players of that group;

The gain is proportional to the invested amount.
By denoting as the payoff of the th player that has chosen the group , which, without loss of generality, is the group where a number of people have privileged information ( with probability ), we have
(1) 
and naturally
(2) 
The reader should observe that the return, in case of a win, is the sum of the investment of all players in the loosing group multiplied by a factor that distributes equally the gain between the participants of the winning group: . It is important to mention that for equal investments, , for all players, the payoff for each player is .
This game is a good example of multiple zerosum game, since the sum of payoff of the players in a match is 0, which means that if one or more players gain a certain amount, the other ones lose this same amout, i.e.,
(3) 
As a partition and the sequences of “bets” are given, respectively, by , and , a simple calculation can be performed to provide us the expected payoff of the th player:
(4) 
and
(5) 
In this regard, it is important to think of a mathematical point of view without taking into consideration the complex interaction that exists among players and how some of them () can use their privileged information to make money. The problem is: giving the partition and the bets and conditioned to this partition, how to compute the average payoff?
For example, if, by hypothesis, we know the probability distribution
that determines how many individuals choose the group as well as the conditional probability distribution function (pdf) that governs the probability of a player which invests in the group to perform a proposal between and , we would have the average payoffs of the individuals in the groups and given, respectively, by(7) 
where and .
At this point, the interactions among players, misinformation, and other relevant features, should lead to more exciting studies for this game since no indication produces the following distributions: and . So, once we showed that our very different approach avoids the Bayesian formalism of the incomplete information game theory Harsanyi , now we are able to define the dynamics for this complex scenario by presenting how the players propagate their privileged information. First, in real situations the information can be propagated with good or bad intentions. Among the players that know the bias in a group (there are players with privileged information), a fraction of them propagate this information with good intention (i.e., saying the truth). On the other hand, there are players which only wish to maximize their profit (sincerity is not their main feature) and will influence all the other players to invest in the other group. In addition, players do not know the chances of each group (or of the corresponding boxer) and therefore, depend on the information from the other players.
So, we have a model with three types of players:

IA – (Informed and altruist players) – They know the probability of a group winning and propagate this information;

IS – (Informed and selfish players) – They know the probability of a group winning but suggests to the misinformed players to invest their contribution in the opposing group lying to them;

U – (uninformed players) – They have no information and depend on the information given by the two first groups.
In addition, the model has four parameters: , , , and , and the profit in each round is given by Eqs. (1) and (2).
Now let us define the dynamics of the game by supposing (without loss of generality) that is the probability of the group defeats the group , as shown in Table 1.
Players  Decision 

I versus I (A or S)  Both players go to the group A 
U versus U  Each player choose a group 
with probability 1/2  
IA versus U  Both players go to the group A 
IS versus U  The IS player goes to the group A 
and the U player goes to the group B 
The group is composed by part of the informed players as well as a fraction of uninformed ones that are convinced by the informed altruist players to invest in this group and half of uninformed players which interact with other uninformed players. On the other hand, the group consists of uninformed players convinced by the informed selfish ones to invest in it and the other half of uninformed players which find other uninformed players.
3 Results
Now we present our main results. In the next subsection we present the most of our results, which is an exploration of the main properties of the game considering that all players invest the same quantity along the time: , which remains fixed over the different iterations of the game. In our numerical approach, we consider in total different players and s is made equal to 1. In the following subsection, we consider an interesting evolutionary aspect of the game, based on reactions of the players according to their gain or loss, so the values , evolve over time but , for all players: .
3.1 Results I: Sampling and probability theory
We first consider a more natural situation where all players invest the same quantity, in every iteration of the game. In this case, we have
According to Table 1, along with the fraction of players, we have
where is the density of informed players, is the density of altruist informed players, and , i.e., the number of players is held constant.
The rate yields
Thus we can write
(8) 
and
(9) 
It is interesting to observe that for , i.e., the informed players have profit in average. This situation leads to
(10) 
This means that is the minimum information level necessary to obtain profit. Another alternative is to think of the level of information (probability of the favourite group to win) and the density of informed players . As we have them in hand, the altruism level required to obtain profit is then given by
(11) 
We can consider applications where , and this deserves a future investigation. But for the sake of simplicity, by thinking in boxers, a draw is very rare, thus we make for all the results obtained from here in this work.
In this work, we perform numerical simulations based on turns. One turn is averaged over different time series corresponding to independent runs. Each time series corresponds to a sequence of iterations (rounds). In each round, pairs of players are randomly matched and all players necessarily participate once (such as performing a matching in a graph). In each round, the players take a decision according to the information and nature of their partners determined by the Table 1. So, for each turn , the average payoff of bettors in the group , calculated via MC simulations, is obtained by a general formulae:
For fixed values of and , one has that is the same for all , i.e., there is no difference among the payoffs of the different players in same group, in the same run of the same turn , since the investments in this version of the game are fixed and made equal to 1 for all players during all the iterations. However, as shown in the next subsection, this is not the truth when the reaction of the players to their profit is taken into consideration.
Figure 1
shows the payoffs obtained by MC simulations. These estimates are compared with our theoretical result predicted by Eqs. (
8) and (9). In Figs. 1 (a), (b), (c), and (d), we show the results for the average payoff of players that have bet in and , for , 3, 10, and 300, respectively. We can observe a good agreement between our numerical and theoretical predictions.We complete our analysis with the plot shown in Fig. 1 (e) which shows the average payoff of the informed and uniformed players. It is important to observe that all informed players bet in the group (or boxer) , but not all player that bets in is an informed player according to our rules. On the other hand every, player that bets in the group is necessarily an uninformed player. Of course, we do not observe difference between the average payoff of the group formed by bettors in the group and the group formed by informed players, however we have that average payoff of the uniformed players group even being greater than that of the informed players group, is smaller than the group formed by the uninformed players that bet in (average over uninformed players conditioned to the group ). This shows that uninformed players which goes to the group are the responsible for this decrease or for getting worse the average of the uninformed players from a general point of view.
By looking into the Fig. 1 (a), (b), (c), and (d), it turns interesting to understand the influence of
in our results, or, in other words, we are wondering how the central limit theorem is working here. When considering small values of
, we have that the payoff switch between two different values. Asenlarges, the superposition among different runs generates a continuous distribution of payoffs which we expect to be a gaussian distribution of payoffs over the different turns.
Figure 2 shows the distribution of payoff over the turns. The plots (a), (b), and (c) show the payoff frequencies for different values of for the bettors in the groups and . We can observe a mixing distribution composed by a gaussian peak and a delta peak migrating to a unique gaussian distribution in both cases: for bettors in and (in monolog scale) as enlarges. The mix distribution occurs because for , each player loses or gains one unit with his group dividing the quantity that changes along different rounds, generating the Gaussian distribution. However it is interesting to observe that the convergence occurs migrating from this bimodal distribution, passing by a multimodal distribution (see for example
) until that such modes are fused in a unique gaussian distribution. The central limit theorem can also be observed by considering the standard deviations over the different turns, calculated as:
(12) 
where it is expected that . As can be seen in Fig. 2 (d), our results are in complete agreement with each other. With this result, we finish our analysis about sampling and agreement between MC simulations and theory.
Now, we explore the relationship between altruism and information in this game. Since the altruism influences the information, we are wondering how the payoff vary for all possible values of information, , and altruism, for a fixed value of , for instance, . Figure 3 shows the payoff of players of the group whereas this is the group of informed players. We can observe isopayoff curves which depend on values of and .
From this figure, one can observe that for low densities of informed players there exist only altruist players with positive payoff since this is a profitable scenario, i.e., highlevel information: . However, for high density of informed players, the pure altruism can bring strong damages for the group and the information is not a trump for the game. In this case, it is interesting to obtain the minimal altruism level required to reach a positive payoff in the scenario predicted by Eq. (11). In Fig. 4, we show the diagram for all possible values of and for . The positive (gray) and negative (purple) payoff regions predicted by MC simulations are regions where each set yields a profit or loss for the players, respectively. The continuous curve (in black) shows the theoretical prediction of the threshold between the profit and loss obtained by Eq. (11).
One can observe a perfect agreement between the simulations and our analytical approach. Finally, we focus our attention on the influence of the information level () on the transition curves versus according to Eq. (11). Figure 5 shows for different values of . As increases, we observe the evolution of the curves . It is interesting to observe that for (this value is obtained by making in Eq. (11), even for a population entirely formed by nonaltruist players, the payoff is always negative. So, our work shows that in a population with people that possess some privileged information, the payoff of the informed players is deeply changed by the altruism level. In addition, there is a critical altruism level which separates the profit from the loss in the payoff of the players. Our analytical results are in complete agreement with MC simulations.
3.2 Some evolutionary aspects: reactive boxers game
Here, we study a simple variation of the game where players may change their investment according to their results. The evolution is not exactly as that one conventionally considered in game theory where the strategies are copied according to, for instance, the best payoff in the exact sense of Darwinian evolutionary theory. In the right case of this paper, we propose a simple evolutionary dynamics where the player only tries to adjust his investment according to his personal losses or gains from a similar way as that used in the ultimatum game studied in Ref. rdasilva2016 or in the public goods game (see for example SilvaIJBC2010 ; SilvaLNE2006 ; SilvaPhysA2006 ). Basically, one considers a parameter used to allow the players to make small adjustments depending on their payoff. If a player has invested at turn of a certain run, and his payoff is positive, then his investment is increased in the next round: . However, if the payoff is negative the investment is decreased in the next round: . So, in this approach, we are able to analyse how the dynamical evolution of the investments affects the wealth of the players.
For this particular analysis it is convenient to consider the cumulative wealth defined as
and, mainly, the average gain per turn: which measures the real average gain.
We add these ingredients to our previous study and repeat the MC simulations to calculate and its standard deviation by considering the different gains among the players in each group. It is important to observe that when compared with the standard deviation calculated in Eq. 12, this measures the variation over the players (internal) while the first measures the variation over the turns (external).
The variance is also averaged over time series. Here, we also choose the same bad situation for the group (, , and ) and start the simulations with all players in the same initial unit investment (, for ). The results for the different groups are presented in Fig. 6. For , the gains remain approximately the same as presented in the beginning of previous subsection since they correspond to similar samplings. However for the situation is disastrous for bettors in the group which is opposite for the bettors in the group that have a considerable increase in average as shown respectively in Figs 6 (a) and (b).
It is important to observe that the variance in the bettors in the group is due to the difference between informed and uninformed ones since the informed bettors necessarily have the same gain at the same turn. However, this does not occur for the bettors in the group , in which the variance is calculated over many different wealths. This is clear in Figs. 6 (c) and (d) presented in scale. We can observe that the variance for the group has an algebraic decay , indicating a diffusive effect, since it means . However, we do not observe this same behaviour for the group , which does not present a power law for . For different values of , the standard deviation first decreases and then, after some time, enlarges which indicates the corroboration with the anomalous behaviour of time evolution of gain.
We also analyse other prosperous situations for the group for , according to diagrams obtained in the last subsection, but the effects of the evolutionary dynamics also destroy their profit, showing that evolutionary dynamics is not good to the informed players differently than situation represented for the group .
4 Summaries and brief conclusions
In conclusion, we proposed a new game which presents an alternative social paradigm in economic scenarios with incomplete information. Our results claim that benefits of information can be destroyed under high altruism, since the informed players, even with high level information (high ), can gain more times but small quantities and the defeats, even rare, are rentless. On the other hand, even earning few times, the uninformed players can outweigh their losses, which are in average more frequent, but inexpressive in absolute value.
In an evolutionary version of the game, we show that the gain of the informed players can get worse if the following approach is adopted: the player increases its investment for positive payoffs, and decreases the investment for negative payoffs.
The results of our analytical approach are corroborated by MC simulations, and the findings suggest that a lot of new applications may be considered in future investigations, including, for instance, the study of spacial effects through the analysis of this game in different networks or even considering the evolutionary aspects on the strategies.
Acknowledgements – The authors would like to thank CNPq (National Council for Scientific and Technological Development) for the partial financial support.
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