Evolution of Coordination in Pairwise and Multi-player Interactions via Prior Commitments
EEvolution of Coordination in Pairwise and Multi-playerInteractions via Prior Commitments
Ogbo Ndidi Bianca , Aiman Elgarig , The Anh Han ,(cid:63) School of Computing, Engineering and Digital Technologies, Teesside University (cid:63)
Corresponding author: The Anh Han ([email protected])
Abstract
Upon starting a collective endeavour, it is important to understand your partners’ preferences and howstrongly they commit to a common goal. Establishing a prior commitment or agreement in terms ofposterior benefits and consequences from those engaging in it provides an important mechanism for se-curing cooperation in both pairwise and multiparty cooperation dilemmas. Resorting to methods fromEvolutionary Game Theory (EGT), here we analyse how prior commitments can also be adopted as a toolfor enhancing coordination when its outcomes exhibit an asymmetric payoff structure, in both pairwiseand multiparty interactions. Arguably, coordination is more complex to achieve than cooperation sincethere might be several desirable collective outcomes in a coordination problem (compared to mutual coop-eration, the only desirable collective outcome in cooperation dilemmas), especially when these outcomesentail asymmetric benefits for those involved. Our analysis, both analytically and via numerical simu-lations, shows that whether prior commitment would be a viable evolutionary mechanism for enhancingcoordination and the overall population social welfare strongly depends on the collective benefit and sever-ity of competition, and more importantly, how asymmetric benefits are resolved in a commitment deal.Moreover, in multiparty interactions, prior commitments prove to be crucial when a high level of groupdiversity is required for optimal coordination. Our results are shown to be robust for different selectionintensities. We frame our model within the context of the technology adoption decision making, but theobtained results are applicable to other coordination problems.
Keywords:
Commitment, Evolutionary Game Theory, Coordination, Technology Adoption.1 a r X i v : . [ c s . M A ] S e p Achieving a collective endeavour among individuals with their own personal interest is an importantsocial and economic challenge in various societies (Hardin, 1968; Ostrom, 1990; Pitt et al., 2012; Barrett,2016; Sigmund, 2010). From coordinating individuals in the workplace to maintaining cooperative andtrust-based relationship among organisations and nations, it success is often jeopardised by individualself-interest (Barrett et al., 2007; Perc et al., 2017). The study of mechanisms that support the evolutionof such collective behaviours has been of great interest in many disciplines, ranging from EvolutionaryBiology, Economics, Physics and Computer Science (Nowak, 2006; Sigmund, 2010; West et al., 2007; Han,2013; Perc et al., 2017; Andras et al., 2018; Kumar et al., 2020). Several mechanisms responsible for theemergence and stability of collective behaviours among such individuals, have been proposed, including kinand group selection, direct and indirect reciprocities, spatial networks, reward and punishment (Nowak,2006; West et al., 2007; Perc et al., 2017; Okada, 2020; Skyrms, 1996).Recently, establishing prior commitments has been proposed as an evolutionarily viable strategy induc-ing cooperative behaviour in the context of pairwise and multi-player cooperation dilemmas (Nesse, 2001;Frank, 1988; Han et al., 2017, 2015a; Sasaki et al., 2015; Arvanitis et al., 2019; Ohtsuki, 2018); namely,the Prisoner’s Dilemma (PD) (Han et al., 2013; Hasan and Raja, 2013) and the Public Goods Game(PGG) (Han et al., 2015a, 2017; Kurzban et al., 2001). It provides an enhancement to different formsof punishment against inappropriate behaviours and of rewards to stimulate the appropriate ones (Chenet al., 2014; Martinez-Vaquero et al., 2015, 2017; Sasaki et al., 2015; Powers et al., 2012; Szolnoki and Perc,2012; Wang et al., 2019), allowing ones to efficiently avoiding free-riders (Han and Lenaerts, 2016; Hanet al., 2015b) and resolving the antisocial punishment problem (Han, 2016). These works have primarilyfocused on modelling prior commitments for improving mutual cooperation among self-interested agents.In the context of cooperation dilemma games (i.e. PD and PGG), mutual cooperation is the only desirablecollective outcome to which all parties are required to commit if an agreement is to be formed. In othercontexts such as coordination problems, this is not the case anymore since there might be multiple optimalor desirable collective outcomes and players might have distinct, incompatible preferences regarding whichoutcome a mutual agreement should aim to achieve (e.g. due to asymmetric benefits). Such coordinationproblems are abundant in nature, ranging from collective hunting and foraging to international climatechange actions and multi-sector coordination (Santos and Pacheco, 2011; Ostrom, 1990; Barrett, 2016;Ohtsuki, 2018; Bianca and Han, 2019; Skyrms, 1996; Santos et al., 2016).Hence, we explore how arranging a prior agreement or commitment can be used as a mechanismfor enhancing coordination and the population social welfare in this type of coordination problems, inboth pairwise and multi-player interaction settings. Before individuals embark on a joint venture, a pre-agreement makes the motives and intentions of all parties involved more transparent, thereby enablingan easier coordination of personal interests (Nesse, 2001; Cohen and Levesque, 1990; Han, 2013; Hanet al., 2015b). Although our approach is applicable for a wide range of coordination problems (e.g.single market product investments as described above), we will frame our models within the technologyinvestment strategic decision making problem, allowing us to describe the models clearly. Namely, wedescribe technology adoption games capturing the competitive market and decision-making process amongfirms adopting new technologies (Zhu and Weyant, 2003; Bardhan et al., 2004), with a key parameter α representing how competitive the market is (thus describing how important coordination is). Similar toprevious commitment models, we will perform theoretical analysis and numerical simulations resortingto stochastic methods from Evolutionary Game Theory (EGT) (Hofbauer and Sigmund, 1998; Sigmundet al., 2010).We will start by modelling a pairwise technology adoption decision making, where two investmentfirms (or players) competing within a same product market who need to make strategic decision on whichtechnology to adopt (Zhu and Weyant, 2003; Chevalier-Roignant et al., 2011), a low-benefit (L) or a high-benefit (H) technology. Individually, adopting H would lead to a larger benefit. However, if both firmsinvest on H they would end up competing with each other leading to a smaller accumulated benefit thanif they could coordinate with each other to choose different technologies. However, given the asymmetryin the benefits in such an outcome, clearly no firm would want to commit to the outcome where its optionis L, unless some form of compensation from the one selecting H can be ensured.We then extend and generalize the pairwise model to a multi-player one, capturing the strategic inter-action between more than two investment firms. In the multi-player model, a key parameter µ is ascribedto the market demand of high technology, i.e. what is the optimal fraction of the firms in a group toadopt H. We analytically examine how players can be coordinated when there is a market demand for aparticular technology. We show that differently from the two-player game, the newly defined parameter µ leads to a new kind of complexity when trying to achieve group coordination. When there is a high levelof diversity in demand (i.e. intermediate values of µ ), as can be seen in different technologies adoptioncontexts (Beede and Young, 1998; Schewe and Stuart, 2015), introducing prior commitment can lead tosignificant improvement in the levels of coordination and population social welfare.The next section discusses related work, which is followed by a description of our models and detailsof the EGT methods for analysing them. Results of the analysis and a final discussion will then follow. The problem of explaining the emergence and stability of collective behaviours has been actively addressedin different disciplines (Nowak, 2006; Sigmund, 2010). Among other mechanisms, such as reciprocity andcostly punishment, closely related to our present model is the study of cooperative behaviours and pre-commitment in cooperation dilemmas, for both two-player and multiplayer games (Han et al., 2013, 2017;Sasaki et al., 2015; Hasan and Raja, 2013). It has shown that to enhance cooperation commitments need tobe sufficiently enforced and the cost of setting up the commitments is justified with respect to the benefitderived from the interactions—both by means of theoretical analysis and of behavioural experiments(Ostrom, 1990; Cherry and McEvoy, 2013; Kurzban et al., 2001; Chen and Komorita, 1994; Arvanitiset al., 2019). Our results show that this same observation is seen for coordination problems. However,arranging commitments for enhancing coordination is more complex, exhibiting a larger behavioural space,and furthermore, their outcomes strongly depend on new factors only appearing in coordination problems;namely, a successful commitment deal needs to take into account the fact that multiple desirable collectiveoutcomes exist for which players have incompatible preferences; and thus how benefits can be sharedthrough compensations in order to resolve the issues of asymmetric benefits, is crucially important (Biancaand Han, 2019).We moved further by expanding our two-player game in the previous work to a multi-player model,the outcome has shown to be more complex as there are more players involved. We yet again investigatedhow coordination and cooperation can be improved using prior commitment deal when there are multipleplayers involved and also when there is a particular market demand (Bianca and Han, 2019). Our approachin exploring how implementing prior commitment enhances cooperation dilemma has also been investigatedby previous researchers in the past (Chen and Komorita, 1994). A good level of cooperation was seen in aPublic Good Game experiment when there was a binding agreement made during the prior communicationstage among members of the group. They hypothesized that if members of a group are allowed to make apledge (a degree of bindings/commitment) before their actual decisions, they will be able to communicatetheir intentions and it will overall increase cooperation rate in the population. As predicted, their resultsclearly demonstrates that making a pledge improves cooperation although the degree of commitmentrequired in the pledge deferentially affected the cooperation rate (Chen and Komorita, 1994; Cherry andMcEvoy, 2013; Kurzban et al., 2001).There have been several other works studying the evolution of coordination, using the so-called StagHunt game, see e.g. (Skyrms, 2003; Pacheco et al., 2009; Santos et al., 2006; Sigmund, 2010). However, tothe best of our knowledge there has been no work studying how prior commitments can be modelled andused for enhancing the outcome of the evolution of coordination. As our results below show, significantenhancement of coordination and population welfare can be achieved via the arrangement of suitablecommitment deals.Furthermore, it is noteworthy that commitments have been studied extensively in Artificial Intelligenceand Multi-agent systems literature, see e.g. (Castelfranchi and Falcone, 2010; Chopra and Singh, 2009;Rzadca et al., 2015). Differently from our work, these studies utilise commitments for the purpose ofregulating individual and collective behaviours, formalising different aspects of commitments (such asnorms and conventions) in multi-agent systems. However, our results and approach provide importantnew insights into the design of such systems as these require commitments to ensure high levels of efficientcollaboration and coordination within a group or team of agents. For example, by providing suitableagreement deals agents can improve the chance that a desirable collective outcome (which is best for thesystems as a whole) is reached even when benefits provided by the outcome are different for the partiesinvolved.
In the following we first describe a two-player technology adoption game then extend it with the option ofarranging prior commitments before playing the game. We then present a multi-player version of the model,with and without commitments, too. Then, we describe the methods, which are based on EvolutionaryGame Theory for finite populations, which will be used to analyse the resulting models.
We consider the scenario that two firms (players) compete for the same product market, and they needto make a (strategic) decision on which technology to invest on, a low-benefit (L) or a high-benefit (H)technology. The outcome of the interaction can be described in terms of costs and benefits of investmentsby the following payoff matrix (for row player): (cid:32)
H LH αb H − c H b H − c H L b L − c L αb L − c L (cid:33) = (cid:32) H LH a bL c d (cid:33) , (1) where c L , c H and b L , b H ( b L ≤ b H ) represent the costs and benefits of investing on L and H, respectively; α ∈ (0 ,
1) indicates the competitive level of the product market: the firms receive a partial benefit ifthey both choose to invest on the same technology. Collectively, the smaller α is (i.e. the higher themarket competitiveness), the more important that the firms coordinate to choose different technologies.For simplicity, the entries of the payoff matrix are denoted by a, b, c, d , as above. We have b > a and c > d . Without loss of generality, we assume that H would generate a greater net benefit, i.e. c = b L − c L < b H − c H = b .Note that although we describe our model in terms of technology adoption decision making, it is gener-ally applicable to many other coordination problems for instance wherever there are strategic investmentdecisions to make (in competitive markets of any products) (Zhu and Weyant, 2003; Chevalier-Roignantet al., 2011). We now extend the model allowing players to have the option to arrange a prior commitment before aTD interaction. A commitment proposal is to ask the co-player to adopt a different technology. That is,a strategist intending to use H (resp., L) would ask the co-player to adopt L (resp., H). We denote thesecommitment proposing strategies as HP and LP, respectively. Similarly to previous models of commitments(for PD and PGG) (Han et al., 2013, 2015a), to make the commitment deal reliable, a proposer pays anarrangement cost (cid:15) . If the co-player agrees with the deal, then the proposer assumes that the opponentwill adopt the agreed choice, yet there is no guarantee that this will actually be the case. Thus whenevera co-player refuses to commit, HP and LP would play H in the game. When the co-player accepts thecommitment though later does not honour it, she has to compensate the honouring co-player at a personalcost δ .Differently from previous models on PD and PGG where an agreed outcome leads to the same payofffor all parties in the agreement (mutual cooperation benefit), in the current model such an outcome wouldlead to different payoffs for those involved. Therefore, as part of the agreement, HP would compensateafter the game an amount θ to accepted player that honours the agreement; while LP would request acompensation θ from such an accepted co-player.Besides HP and LP, we consider a minimal model with the following (basic) strategies in this commit-ment version: • Non-proposing acceptors, HC and LC, who always commit when being proposed a commitment dealwherein they are willing to adopt any technology proposed (even when it is different from theirintended choice), honour the adopted agreement, but do not propose a commitment themselves.They play their intended choice, i.e. H and L, respectively, when there is no agreement in place; • Non-acceptors, HN and LN, who do not accept commitment, play their intended choice during thegame, and do not propose commitments; • Fake committers, HF and LF, who accept a commitment proposal yet play the choice opposite towhat has been agreed whenever the game takes place. These players assume that they can exploitthe commitment proposing players without suffering the consequences .Note that similar to the commitment models for the PD game (Han et al., 2013), some possible strategieshave been excluded from the analysis since they are dominated by at least one of the strategies in anyconfiguration of the game: they can be omitted without changing the outcome of the analysis. Forexample, those who propose a commitment (i.e. paying a cost (cid:15) ) but then do not honour (thus haveto pay the compensation when facing a honouring acceptors) would be dominated by the correspondingnon-proposers.Together the model consists of eight strategies that define the following payoff matrix, capturing theaverage payoffs that each strategy will receive upon interaction with one of the other seven strategies(where we denote λ = θ + θ , λ = b − (cid:15) − θ , λ = c − (cid:15) + θ , λ = a − (cid:15) + δ and λ = d − (cid:15) + δ , just forthe sake of clear representation) HP LP HN LN HC LC HF LFHP b + c − (cid:15) b − (cid:15) − λ a b λ λ λ λ LP c − (cid:15) + λ b + c − (cid:15) a b λ λ λ λ HN a a a b a b a b LN c c c d c d c d HC c + θ b − θ a b a b a b LC c + θ b − θ c d c d c d HF a − δ d − δ a b a b a b LF a − δ d − δ c d c d c d . (2) Note that when two commitment proposers interact only one of them will need to pay the cost of setting upthe commitment. Yet, as either one of them can take this action they pay this cost only half of the time (onaverage). In addition, the average payoff of HP when interacting with LP is given by ( b − (cid:15) − θ + b − θ ) = (2 b − (cid:15) − θ − θ ). When two HP players interact, each receives ( b − (cid:15) − θ + c + θ ) = ( b + c − (cid:15) ). Compared to cooperation dilemmas such as PD and PGG, fake strategies make less sense in the context of coordinationgames since they would not earn the temptation payoff by adopting a different choice from what being agreed. Moreover, inthe presence of an agreement, players obtain an additional compensation when adopting the disadvantageous choice (i.e. L).We will keep the fake strategies in the analysis of pairwise games for confirmation of these intuitions but will exclude themfrom multi-player settings for simplicity, without being detrimental to the results.
We say that an agreement is fair if both parties obtain the same benefit when they honour it (afterhaving taken into account the cost of setting up the agreement). For that, we can show that θ and θ mustsatisfy θ = b − c − (cid:15) and θ = b − c + (cid:15) , and thus, both parties obtain b + c − (cid:15) . Indeed, they can be achievedby comparing the payoffs of HP and HC when they interact, i.e. b − c − θ = c + θ , where solving thisequation we would obtain θ = b − c − (cid:15) .With these conditions it also ensures that the payoffs of HP and LP when interacting with each otherare equal. Our analysis below will first focus on whether and when the fair agreements can lead toimprovement in terms of coordination and the overall social welfare (i.e. average population payoff). Wewill discuss how different kinds of agreements (varying θ and θ ) affect the outcome, with additionalresults provided in Appendix. We now describe a N -player ( N >
2) version of the TD model. Again, as before, we will introduce themodel in the context of technology investment market decision making. In a group (of size N ) with k players of type H (i.e., N − k players of type L ), the expected payoffs of playing H and L can be writtenas follows Π H ( k ) = α H ( k ) b H − c H , Π L ( k ) = α L ( k ) b L − c L , (3)where α H ( k ) and α L ( k ) represent the fraction of the benefit obtained by H and L players, respectively,which depend on the composition of the group, k . For two-player TD, both are equal to α . To generalizefor N -player TD interactions, they should also depend on the demand for high technology (H) in thegroup, describing what is the maximal number of players in the group that can adopt H without reducingtheir benefit due to competition. Let us denote this number by µ (where 1 ≤ µ ≤ N ). For example,intermediate values of µ indicate a high level of group diversity is needed for optimal coordination. When µ = N , it means there is a significant market demand of the high benefit technology so that all firms canadopt it without leading to competition.Hence, we define α H ( k ) = , if k ≤ µ, α µk otherwise , (4) α L ( k ) = , if k ≥ µ, α ( N − µ ) N − k otherwise . (5)The rationale of these definitions is that whenever k ≤ µ , full benefits from adopting H can be obtained,and moreover, if k > µ , the larger k the stronger the competition is among H adopters. Similarly for Ladopters. The parameters α and α stand for the intensities of competition for investing in H and in L,respectively. For simplicity we assume in this paper α = α = α . Note that for N = 2 we recover thetwo-player model given in Equation (1).The optimal group payoff is achieved when there are exactly µ players adopting H and the rest adoptingL, leading to an average payoff for each member given by A := µ ( b H − c H ) + ( N − µ )( b L − c L ) N .
We can define the N -player game version with prior commitments in a similar fashion as in the two-playergame. Commitment proposing strategists (i.e. HP and LP players) will propose before an interactionthat the group will play the optimal arrangement (so that every player obtains an average payoff A ). Forsimplicity, we assume that the committed players adopt the fair agreement, i.e. every member will obtainthe same payoff after compensation is made to those adopting L. As such, we don’t need to consider whowill adopt H or L, as all would receive the same payoff at the end. Moreover, whenever a player in thegroup refuses to commit, commitment proposers will adopt H. Details of payoff calculation will be providedin Results section (cf. Table 1). In this work, we will perform theoretical analysis and numerical simulations (see next section) using EGTmethods for finite populations (Nowak et al., 2004; Imhof et al., 2005). Let Z be the size of the population.In such a setting, individuals’ payoff represents their fitness or social success , and evolutionary dynamicsis shaped by social learning (Hofbauer and Sigmund, 1998; Sigmund, 2010), whereby the most successfulindividuals will tend to be imitated more often by the other individuals. In the current work, social learningis modelled using the so-called pairwise comparison rule (Traulsen et al., 2006), a standard approach inEGT, assuming that an individual A with fitness f A adopts the strategy of another individual B withfitness f B with probability p given by the Fermi function, p A,B = (cid:16) e − β ( f B − f A ) (cid:17) − . The parameter β represents the ‘imitation strength’ or ‘intensity of selection’, i.e., how strongly the in-dividuals base their decision to imitate on fitness difference between themselves and the opponents. For β = 0, we obtain the limit of neutral drift – the imitation decision is random. For large β , imitationbecomes increasingly deterministic.In the absence of mutations or exploration, the end states of evolution are inevitably monomorphic:once such a state is reached, it cannot be escaped through imitation. We thus further assume that, witha certain mutation probability, an individual switches randomly to a different strategy without imitatinganother individual. In the limit of small mutation rates, the dynamics will proceed with, at most, twostrategies in the population, such that the behavioural dynamics can be conveniently described by a MarkovChain, where each state represents a monomorphic population, whereas the transition probabilities aregiven by the fixation probability of a single mutant (Imhof et al., 2005; Nowak et al., 2004). The resultingMarkov Chain has a stationary distribution, which characterises the average time the population spendsin each of these monomorphic end states.Before describing how to calculate this stationary distribution, we need to show how payoffs are calcu-lated, which differ for two-player and N-player settings, as below. • Average Payoff for the Two Player Game
Let π ij represent the payoff obtained by strategist i in each pairwise interaction with strategist j ,as defined in the payoff matrices in Equations (1) and (2). Suppose there are at most two strategiesin the population, say, k individuals using i (0 ≤ k ≤ Z ) and ( Z − k ) individuals using j . Thus theaverage payoff of the individual that uses i or j can be written respectively as followsΠ i ( k ) = ( k − π ii + ( Z − k ) π i,j Z − , Π j ( k ) = kπ j,i + ( Z − k − π j,j Z − . (6) • Expected Payoff in The Multiplayer Game
In the case of N -player interactions, suppose the population includes x individuals of type i and Z − x individuals of type j . The probability to select k individuals of type i and N − k individuals oftype j , in N trails, is given by the hypergeometric distribution as follows (Sigmund, 2010; Gokhaleand Traulsen, 2010) H ( k, N, x, N ) = (cid:0) xk (cid:1)(cid:0) Z − xN − k (cid:1)(cid:0) ZN (cid:1) Hence, in a population of x i -strategists and ( Z − x ) j strategists, the average payoff of i and j aregiven by Π ij ( k ) = N − (cid:88) k =0 H ( k, N − , x − , Z − π ij ( k + 1) = N − (cid:88) k =0 (cid:0) x − k (cid:1)(cid:0) Z − xN − − k (cid:1)(cid:0) Z − N − (cid:1) π ij ( k + 1) , Π ji ( k ) = N − (cid:88) k =0 H ( k, N − , x − , Z − π ji ( k ) = N − (cid:88) k =0 (cid:0) xk (cid:1)(cid:0) Z − − xN − − k (cid:1)(cid:0) Z − N − (cid:1) π ij ( k ) . (7)Now, for both two-player and N -player settings, the probability to change the number k of individualsusing strategy A by ± one in each time step can be written as (Traulsen et al., 2006) T ± ( k ) = Z − kZ kZ (cid:104) e ∓ β [Π i ( k ) − Π j ( k )] (cid:105) − . (8)The fixation probability of a single mutant with a strategy i in a population of ( N −
1) individuals using0 j is given by (Traulsen et al., 2006; Nowak et al., 2004) ρ j,i = N − (cid:88) i =1 i (cid:89) j =1 T − ( j ) T + ( j ) − . (9)Considering a set { , ..., q } of different strategies, these fixation probabilities determine a transition matrix M = { T ij } qi,j =1 , with T ij,j (cid:54) = i = ρ ji / ( q −
1) and T ii = 1 − (cid:80) qj =1 ,j (cid:54) = i T ij , of a Markov Chain. The normalisedeigenvector associated with the eigenvalue 1 of the transposed of M provides the stationary distributiondescribed above (Imhof et al., 2005), describing the relative time the population spends adopting each ofthe strategies. Risk-dominance
An important measure to determine the evolutionary dynamic of a given strategy isits risk-dominance against others. For the two strategies i and j , risk-dominance is a criterion whichdetermine which selection direction is more probable: an i mutant is able to fixating in a homogeneouspopulation of agents using j or a j mutant fixating in a homogeneous population of individuals playing i .In the case, for instance, the first was more probable than the latter then we say that i is risk-dominant against j (Nowak et al., 2004; Sigmund, 2010) which holds for any intensity of selection and in the limitfor large population size Z when N (cid:88) k =1 Π i,j ( k ) ≥ N − (cid:88) k =0 Π j,i ( k ) (10)This condition is applicable for both two-player games, N = 2, and when N-player games with N >
We will first describe results for two-player games, then proceeding to provide those for the N -playerversion. To begin with, using the conditions given in Equation 10, we obtain that if θ + θ < b − c then HP is risk-dominant (see Methods) against LP. Otherwise, LP is risk-dominant against HP.Similarly, we derive the conditions regarding the commitment parameters for which HP and LP areevolutionarily viable strategies, i.e. when they are risk-dominant against all other non-proposing ones.Indeed, HP and LP are risk-dominant against all other six non-proposing strategies, respectively, if and1only if (cid:15) < min { b + c − a, b − c − d, b − c − a − θ , b − c − d − θ , b + c − a + 4 δ , b + c − d + 4 δ } ,(cid:15) < min { b + c − a, b − c − d, c − b − a + 4 θ , c − b − d + 4 θ , b + c − a + 4 δ , b + c − d + 4 δ } . (11)Note that each element in the min expressions above corresponds to the condition for one of the sixnon-proposing strategies HN, LN, HC, LC, HF, LF, respectively.Thus, we can derive the conditions for θ , θ and δ : θ <
14 (3 b − c − (cid:15) − { a, d } ) ,θ >
14 ( b − c + 3 (cid:15) + 2 max { a, d } ) ,δ >
14 (3 (cid:15) − b − c + 2 max { a, d } ) . (12)In particular, for fair agreements, i.e. θ = ( b − c − (cid:15) ) / θ = ( b − c + (cid:15) ) /
2, we obtain (cid:15) < b + c − { a, d } ,δ >
14 (3 (cid:15) − b − c + 2 max { a, d } ) . (13)It is because 3 b − c − d > b + c − { a, d } , which is due to b > c and max { a, d } ≥ d .In general, these conditions indicate that for commitments to be a viable option for improving coordi-nation, the cost of arrangement (cid:15) must be sufficiently small while the compensation associated with thecontract needs to be sufficiently large (see already Figure 2 for numerical validation). Furthermore, for thefirst condition to hold, it is necessary that b + c > { a, d } . It means that the total payoff of two playerswhen playing the TD game is always greater when they can coordinate to choose different technologies,than when they both choose the same technology.Moreover, the conditions in Equation 13 can be expressed in terms of α and the costs and benefits ofinvestment, as follows (see again the payoff matrices in Equation 1) α < { c H + b L − c L − (cid:15) b H , c L + b H − c H − (cid:15) b L } ,α < { c H + b L − c L − (cid:15) + 4 δ b H , c L + b H − c H − (cid:15) + 4 δ b L } , which can be rewritten as α < { c H + b L − c L − max { (cid:15), (cid:15) − δ } b H , c L + b H − c H − max { (cid:15), (cid:15) − δ } b L } . (14)2This condition indicates under what condition of the market competitiveness and the costs and benefits ofinvesting in available technologies, commitments can be an evolutionarily viable mechanism. Intuitively,for given costs and benefits of investment (i.e. fixing c L , c H , b L , b H ), a larger cost of arranging a (reliable)agreement, (cid:15) , leads to a smaller threshold of α where commitment is viable. Moreover, given a commitmentsystem (i.e. fixing (cid:15) and δ ), assuming similar costs of investment for the two technologies, then a largerratio of the benefits obtained from the two technologies, b H /b L , leads to a smaller upper bound for α forwhich commitment is viable.Remarkably, our numerical analysis below (see already Figure 1) shows that the condition in Equation14 accurately predicts the threshold of α where commitment proposing strategies (i.e. HP and LP) arehighly abundant in the population, leading to improvement in terms of the average population payoffcompared to when commitment is absent (Figure 3).On the other hand, when α is sufficiently large, little improvement can be achieved, especially when b H /b L is large (which is in accordance with the analytical results above). We calculate the stationary distribution in a population of eight strategies, HP, LP, HN, LN, HC, LC,HF and LF, using methods described above. In Figure 1, we show the frequency of these strategies as afunction of α , for different values of (cid:15) and game configurations. In general, the commitment proposingstrategies HP and LP dominate the population when α is small while HN and HC dominate when α is sufficiently large even with different values of beta utilized in the comparison. That is, commitmentproposing strategies are viable and successful whenever the market competitiveness is high, leading to theneed of efficient coordination among the competing players/firms to ensure high benefits. Notably, weobserve that the thresholds of α below which HP and LP are dominant, closely corroborate the analyticalcondition described in Equation 14, in all cases. This observation is also robust for different values ofintensity of selection, β .This observation is robust for varying commitment parameters, i.e. the cost of arranging commitment, (cid:15) , and the compensation cost associated with commitment, δ , see Figure 2. Namely, we show the totalfrequency of commitment strategies (i.e. sum of the frequencies of HP and LP) for varying these parametersand for different values of α . It can be seen that, in general, the commitment strategies dominate thepopulation whenever (cid:15) is sufficiently small and δ is sufficiently large. This observation is in accordancewith previous commitment modelling works for the cooperation dilemma games (Han et al., 2013, 2015a,2017). In addition, we observe that in the current coordination problem, that the smaller α is, thesecommitment strategies dominate the population for wider range of (cid:15) and δ . Our additional results showthat these observations are robust with respect to other game configurations. Furthermore, the resultsshow that increasing β only have some effect for large α , where sharp increase in commitment frequencywhen δ is sufficiently larger.Now, in order to determine whether and when commitments can actually lead to meaningful improve-ment, in Figure 3, we compare the average population payoff or social welfare when a commitment ispresent and when it is absent. In general, it can be seen that when α is sufficiently small (below a thresh-3 HP LP HN LN HC LC HF LF ?1 =1.95, ?2 =2.05 ?1 =1.5, ?2 =2.5 ?1 =1, ?2 =3b ce b =5, c =1d fa ihg ? =0.1 ? =0.01 ? =2 ? =0.01 ? =1 ? =0.01 ? =1 ? =1 ? =0.1 ? =1 ? =2 ? =0.1 ? =1 ? =0.1 ? =0.1 ? =0.1 ? =2 ? =1 Figure 1:
Frequency of the eight strategies, HP, LP, HN, LN, HC, LC, HF and LF, as a functionof α , for different values of (cid:15) and β . In general, the commitment proposing strategies HP and LP dominate thepopulation when α is small while HN and HC dominate when α is sufficiently large in all cases, which is robust fordifferent values of intensity of selection, β . The HN and HC dominate the population as the market competitiondecreases (i.e. when α increases). Larger values of β increase the difference between strategies’ frequencies but donot change the outcomes in general. Parameters: in all panels c H = 1, c L = 1, b L = 2 (i.e. c = 1), b H = 6 (i.e. b = 5). Other parameters: δ = 6; β = 0 . , . Z = 100; Fair agreements are used, where θ and θ are given by θ = ( b − c − (cid:15) ) / θ = ( b − c + (cid:15) ) / a? =0.1 ? =0.01 g? =0.1 ? =1 d? =0.1 ? =0.1 e? =0.5 ? =0.1 f? =0.9 ? =0.1 b? =0.5 ? =0.01 c? =0.9 ? =0.01 i? =0.9 ? =1 h? =0.5 ? =1 Figure 2:
Total frequency of commitment strategies (i.e. sum of the frequencies of HP and LP), as afunction of (cid:15) and δ , for different values of α and β . Primarily, the commitment proposing strategies dominate thepopulation whenever (cid:15) is sufficiently small and δ is sufficiently large. Furthermore, the smaller α , these commitmentstrategies dominate for a wider range of (cid:15) and δ , especially when α is smaller. Increasing β only have some effectfor large α , where sharp increase in commitment frequency when δ is sufficiently larger. Parameters: in all panels c H = 1, c L = 1, b L = 2 (i.e. c = 1), and b H = 6 (i.e. b = 5). Other parameters: β = 0 .
01 in the first, β = 0 . β = 1 in the third row; population size Z = 100; Fair agreements are used, where θ and θ aregiven by θ = ( b − c − (cid:15) ) / θ = ( b − c + (cid:15) ) / f ? =1 d ? =0.01 e ? =0.1 a ? =0.01 b =2, c =1 b ? =0.1 c ? =1 b =5, c =1 W ith Com m itm ent , ? =0.1 W ith Com m itm ent, ? =1 W ith Com m itm ent, ? =2 W ithout Com m itm ent
Figure 3:
Average population payoff as a function of α , when commitment is absent and when itis present, for different values of (cid:15) and β . We observed that when α is small, significant improvement interms of the average population payoff can be achieved through prior commitment. When α is sufficiently large,commitment leads to on improvement or might even be detrimental for social welfare, especially when β is small.That is, at α = 0 . α = 0 . c H = 1, c L = 1, b L = 2 (i.e. c = 1); in panel a, b and c) b H = 6 (i.e. b = 5) with β = 0 . , . and b H = 3 (i.e. b = 2) with β = 0 . , . and δ = 6; population size Z = 100; Fair agreements are used, where θ and θ are given by θ = ( b − c − (cid:15) ) / θ = ( b − c + (cid:15) ) / (cid:15) , the greater improvementis obtained. When α is sufficiently large, commitment leads to on improvement or might even be detri-mental for social welfare, especially when b H /b L is large (which is in accordance with the analytical resultsabove). The detriment is further increased when β is small. We can observe that the thresholds for whicha notable improvement can be achieved is the same as the one for the viability of HP and LP (i.e. asdescribed in Equation 14). N -player TD game As mentioned above, compared to cooperation dilemmas such as PD and PGG, fake strategies make lesssense in the context of coordination games since they would not earn the temptation payoff by adopting adifferent choice from what being agreed. To focus on the group effect and the effect of the newly introducedparameter µ , we will consider a population consisting of HP, LP, HN, LN, HC and LC (i.e. excluding fakestrategies). As shown in the two-player game analysis, the fake strategies (i.e. HF and LF) are not viableoptions in TD games and can be ignored. It is equivalent to consider to the full set of strategies with asufficiently large δ .First of all, we derive the payoffs received by each strategy when encountering specific other strategies(see a summary in Table 1). Namely, Π ij ( k ) and Π ji ( k ) denote the payoffs of a strategist of type i and j , respectively, in a group consisting of k player of type i and N − k players of type j . The first columnof the table lists all possible strategies which can be used by player i (focal player), where as the secondcolumn shows strategies of co-players (opponents). The third column shows the payoffs of focal players. We now derive the conditions under which HP is risk-dominant against the rest of strategies. Since weassume fair agreements, the conditions for LP would be equivalent to those for HP in terms of risk-dominance.First, H P is risk-dominant against HC if N (cid:88) k =1 Π HP,HC ( k ) ≥ N − (cid:88) k =0 Π HC,HP ( k ) , which can be written as N (cid:88) k =1 (cid:16) A − (cid:15)k (cid:17) ≥ Π H ( N ) + N − (cid:88) k =1 A, Hence we obtain (cid:15) ≤ A − Π H ( N ) H N , (15)7Focal Player ( i ) Opponent ( j ) Π i,j ( k )HP, LP HP, LP A − (cid:15)/N HP, LP HC, LC A − (cid:15)/k HP, LP HN Π H ( N ) (for k < N )HP, LP LN Π H ( k ) (for k < N )HN HP,LP, HN,HC Π H ( N )HN LN,LC Π H ( k )LN HP, HN,HC Π L ( k )LN LN,LC Π L ( N )LN LP Π L ( k )HC, LC HP,LP A (for k < N )HC HN,HC Π H ( N )HC LN,LC Π H ( k )LC HN,HC Π L ( k )LC LN,LC Π L ( N ) Table 1: Average payoffs of focal strategy i when facing strategy j , in a group of k former and N − k latterstrategists. H N = (cid:80) Nk =1 1 k .Similarly, HP is risk-dominant against LC if (cid:15) ≤ A − Π L ( N ) H N , (16)For risk-dominance of HP against HN, N (cid:88) k =1 Π HP,HN ( k ) ≥ N − (cid:88) k =0 Π HN,HP ( k ) , which equivalently can be written as A − (cid:15)N ≥ Π H ( N ) , or, (cid:15) ≤ N (cid:0) A − Π H ( N ) (cid:1) . (17)Finally, HP is risk-dominant against LN if N (cid:88) k =1 Π HP,LN ( k ) ≥ N − (cid:88) k =0 Π LN,HP ( k ) , which can be rewritten as A − (cid:15)N + N − (cid:88) k =1 Π H ( k ) ≥ N − (cid:88) k =0 Π L ( k )Further simplification leads to (cid:15) ≤ N A + µb H + αµb H N − (cid:88) k = µ +1 k − ( N − c H − (cid:34) α ( N − µ ) b L µ (cid:88) k =0 N − k + ( N − µ − b L − N c L (cid:35) . (18)In short, in order for commitment proposers to be risk-dominant against all other strategies, it requiresthat (cid:15) is sufficiently small, namely, smaller than minimum of the right hand sides of Equations (15)-(18).9 b =2, c =1 b =5, c =1 HP LP HN HC LCLN e µ =2 d µ =1 f µ =5 a µ =1 b µ =2 c µ =5 Figure 4:
Frequency of the six strategies HP, LP, HN, LN, HC and LC, as a function of (cid:15) in a N-playergame with commitment, for different values of µ . In the N-player game, the new parameter µ describes themarket demand for a high technology, which was set to 1 in the pairwise game. HP and LP have a high frequency forsufficiently small (cid:15) for µ = 2 in both games, and also when µ = 1 for the first, easy coordinate situation (first row).When µ = 5, i.e. when all players can adopt H without benefit reduction, HC always dominate and commitmentstrategies are not successful. This means that when there is a need for a diversity of technology adoption, initiatingprior commitments to enhance coordination is important. Parameters: in panel a, b and c) b H = 6 (i.e. b = 5)with µ = 1 , , b H = 3 (i.e. b = 2) with µ = 1 , , N = 5, β = 0 . α = 0 . c H = 1, c L = 1, b L = 2 (i.e. c = 1); . We compute stationary distributions in a population of six strategies HP, LP, HN, LN, HC and LC, forthe N-player TD game, using the payoffs in Table 1 and the Methods described above. To begin with,in Figure 4 (see also Figure 9 in Appendix), we provide numerical validation for the analytical conditionsobtained in the previous section regarding when commitment proposing strategies are evolutionarily viablestrategies (being risk-dominant against others). Similar to the pairwise TD game, we observe that thereis a threshold for (cid:15) below which it is the case. Moreover, Figure 5 shows that the frequencies of thesestrategies (HP and LP) decrease for increasing α . They dominate the population whenever (cid:15) is sufficientlysmall (e.g. (cid:15) = 0 . a ? =0.1 b =2, c =1 b =5, c =1 HP LP HN HC LCLN b ? =1 c ? =2 f ? =2 e ? =1 d ? =0.1 Figure 5:
Frequency of the six strategies HP, LP, HN, LN, HC and LC, as a function of α in amultiplayer game with commitment , for different values of (cid:15) and also two different game configurations. Ingeneral, the commitment proposing strategies (HP and LP) decrease in frequency for increasing α . They dominateover other strategies for sufficiently small α and (cid:15) . That is, it is more beneficial to engage in a prior commitmentdeal when the market competition is fierce and the cost of arranging the commitment is very minimal. Parameters:in all panels c H = 1, c L = 1, b L = 2 (i.e. c = 1); in panel a, b and c) b H = 6 (i.e. b = 5) with (cid:15) = 0 . , b H = 3 (i.e. b = 2) with (cid:15) = 0 . , N = 5, β = 0 . µ = 2. b =5, c =1 b =2, c =1ba Figure 6:
Total frequency of commitment proposing strategies HP and LP as a function of µ and (cid:15) .In general, the commitment proposing strategies are most successful for intermediate values of µ , especially for asufficiently small cost of arranging prior commitment (cid:15) . Parameters: in all panels, c H = 1, c L = 1 (i.e. c = 1), b L = 2. In panel a), b H = 6 (i.e. b = 5) and in panel b) b H = 3 (i.e. b = 2). Other parameters: N = 5, β = 0 . α = 0 . market competition is harsher (i.e. small α ). These results are robust for different intensities of selection(see Figure 10 in Appendix). In general, our results confirm the similar observations regarding the effectsof (cid:15) and α on the evolutionary outcomes to obtained in the pairwise game above.We now focus on understanding the effect of the new parameter in the N-player game, µ , on theevolutionary outcomes. Recall that µ indicates the demand for high technology (H) in the group, describingwhat is the maximal number of players in the group that can adopt H without reducing their benefit due tocompetition. Figure 4 shows the effect of different values of µ on the frequency or evolutionary success of allstrategies as a function of (cid:15) . When µ is small to intermediate, and the cost of arranging prior commitmentis also small, the commitment proposing strategies are dominant. This suggests that arranging priorcommitments might be more beneficial in such instances. These results also imply that µ is very essentialin determining when commitment should be initiated. Apparently, the greater need for a group mixtureor market diversity of technologies, indicating a more difficult coordination situation, the greater need forthe utilization of commitment to enhance coordination among players is. This observation is even moreevident in Figure 6, where we examine the success of commitment for varying µ and (cid:15) , in regards to twodifferent game configurations. It can be observed that an intermediate value of µ leads to the highestfrequency of commitment strategies, especially in the more difficult coordination situation (i.e. the rightpanel).We now closely examine the gain in terms of social welfare improvement when using prior commitments.As shown in Figure 7, whenever µ < N ( N = 5), i.e. there is a need to coordinate among the group players2 W ith Com m itm ent , ? =0.1 W ith Com m itm ent, ? =1 W ith Com m itm ent, ? =2 W ithout Com m itm ent
Figure 7:
Average population payoff (social welfare) as a function of µ with different values of (cid:15) ,showing when commitment is absent against when it is present. We compare results for different valuesof β in two game configurations. We observe that whenever µ < b H = 6 (i.e. b = 5), in panel d,e and f) b H = 3 (i.e. b = 2). Other parameters: N = 5, α = 0 . c H = 1, c L = 1, b L = 2 µ and higher values of intensity of selection, β . We have described in this paper novel evolutionary game theory models showing how prior commitmentscan be adopted as an efficient mechanism for enhancing coordination, in both pairwise and multi-playerinteractions. For that, we described technology adoption (TD) games where technology investment firmswould achieve the best collective outcome if they can coordinate with each other to adopt a mixture ofdifferent technologies. To this end, a parameter α was used to capture the competitiveness level of theproduct market and how beneficial it is to achieve coordination, while another parameter µ to capturethe optimal coordination mixture or diversity of technology adopters in a group (in the pairwise case, weassume the optimal mixture is where two firms adopt different technologies to avoid conflict).In the coordination settings, there are multiple desirable outcomes and players have distinct preferencesin terms of which outcome should be agreed upon, thus leading to a larger behavioural space than in thecontext of cooperation dilemmas (Han et al., 2013, 2017, 2015a; Sasaki et al., 2015; Hasan and Raja,2013). We have shown that whether commitment is a viable mechanism for promoting the evolution ofcoordination, strongly depends on α : when α is sufficiently small, prior commitment is highly abundantleading to significant improvement in terms of social welfare (i.e. population avarage payoff), comparedto when commitment is absent. Importantly, we have derived the analytical condition for the threshold of α below which the success of commitments is guaranteed, for both pairwise and multi-player TD games.Furthermore, moving from pairwise to a multi-player setting, it was shown that µ plays an important rolefor the success of commitment strategies as well. In general, when µ is intermediate, equivalent to a highlevel of diversity in group choices, arranging prior commitments proved to be highly important. It led tosignificant improvement in terms of social welfare, especially in a harsher coordination situation.In the main text, we have considered that a fair agreement is arranged. In the Appendix (Figure 8), wehave shown that whenever commitment proposers are allowed to freely choose which deal to propose to theirco-players, our results show that, in a highly competitive market (i.e. small α ), commitment proposersshould be strict (i.e. sharing less benefits), while when the market is less competitive, commitmentproposers should be more generous.In short, our analysis has demonstrated that commitment is a viable tool for promoting the evolution ofdiverse collective behaviours among self-interested individuals, beyond the context of cooperation dilemmaswhere there is only one desirable collective outcome (Nesse, 2001; Skyrms, 1996; Barrett et al., 2007). Infuture work, we will consider how commitments can solve more complex collective problems, e.g. in atechnological innovation race (Han et al., 2019), bargaining games (Zisis et al., 2015; Rand et al., 2013),climate change actions (Barrett et al., 2007) and cross-sector coordination (Santos et al., 2016), where theremight be a large number of desirable outcomes or equilibriums, especially when the number of players in4an interaction increases (Duong and Han, 2015; Gokhale and Traulsen, 2010; Han et al., 2012). T.A.H and A.E. are supported by Future of Life Institute (grant RFP2-154).5 θ and θ In the main text, we assume that a fair agreement is always arranged. We consider here what wouldhappen if HP and LP can personalise the commitment deal they want to propose, i.e. any θ and θ canbe proposed (instead of always being fair). Namely, Figure 8 shows the average population payoff varyingthese parameters, for different values of α . We observe that when α is small, the highest average payoff isachieved when θ is sufficiently small and θ is sufficiently large, while for large α , it is reverse for the twoparameters. That is, in a highly competitive market (i.e. small α ), commitment proposers should be strict(HP keeps sufficient benefit while LP requests sufficient payment, from their commitment partners), whilewhen the market is less competitive (i.e. large α ), commitment proposers should be more generous (HPproposes to give a larger benefit while LP requests a smaller payment, from their commitment partners).Our results confirm that this observation is robust for different values of (cid:15) , δ and β . See Figure 9 for numerical results confirming the risk-dominant conditions in the N-player game in themain text.6 a? =0.1 ? =0.01 g? =0.1 ? =1 d? =0.1 ? =0.1 e? =0.5 ? =0.1 f? =0.9 ? =0.1 b? =0.5 ? =0.01 c? =0.9 ? =0.01 i? =0.9 ? =1 h? =0.5 ? =1 Figure 8:
Average population payoff as a function of θ and θ , for different values of α and β . When α is small (panels a and b), the highest average payoff is achieved when θ is sufficiently small and θ is sufficientlylarge, while for large α (panel c), it is the case when θ is sufficiently large and θ is sufficiently small. Figure 4 alsoshows that for a small value of β , the highest average payoff is achieved when α is very minimal compared to otherpanels with higher value of β (compare panel a, d and g). Parameters: in all panels c H = 1, c L = 1, b L = 2 (i.e. c = 1), and b H = 6 (i.e. b = 5). Other parameters: δ = 4 , (cid:15) = 1; β = 0 . , . Z = 100. b =5, c =1, µ =2, ? =0.5 Figure 9:
Validation for the analytical conditions under which HP is risk dominant against strategyHC, LC, HN and LN, see main text.
In all cases, with a small value of (cid:15) , the HP strategy dominated otherplayers. This result of this figure is in accordance with our equations derived above. Parameters: in all panels, c H = 1, c L = 1, b H = 6 (i.e. b = 5), µ = 2 and α = 0 . be b =5, c =1 d fa i h ? =0.1 ? =0.01 ? =2 ? =0.01 ? =1 ? =0.01 ? =2 ? =0.1 HP LP HN HC LC c ? =2 ? =0.01 d ? =0.1 ? =0.1 ? =1 ? =0.1 ? =2 ? =1 g h ? =0.1 ? =1 ? =1 ? =1 LN Figure 10:
Frequency of six strategies HP, LP, HN, LN, HC and LC, as a function of α and fordifferent values of (cid:15) and β . The commitment proposing strategies HP and LP dominate the population whenthe values of α and (cid:15) are sufficiently small, in all cases of β . Furthermore, as the value of (cid:15) increases, the non-proposing strategies dominate the population. Parameters: in all panels c H = 1, c L = 1, b L = 2 (i.e. c = 1), b H = 6 (i.e. b = 5); Other parameters: (cid:15) = 0 . , , β = 0 . , . , References
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