# Evolution of Coordination in Pairwise and Multi-player Interactions via
Prior Commitments

Ndidi Bianca Ogbo1, Aiman Elgarig1, The Anh Han1,⋆

1School of Computing, Engineering and Digital Technologies, Teesside
University
⋆ Corresponding author: The Anh Han<EMAIL_ADDRESS>
## Abstract

Upon starting a collective endeavour, it is important to understand your
partners’ preferences and how strongly they commit to a common goal.
Establishing a prior commitment or agreement in terms of posterior benefits
and consequences from those engaging in it provides an important mechanism for
securing cooperation. Resorting to methods from Evolutionary Game Theory
(EGT), here we analyse how prior commitments can also be adopted as a tool for
enhancing coordination when its outcomes exhibit an asymmetric payoff
structure, in both pairwise and multiparty interactions. Arguably,
coordination is more complex to achieve than cooperation since there might be
several desirable collective outcomes in a coordination problem (compared to
mutual cooperation, the only desirable collective outcome in cooperation
dilemmas). Our analysis, both analytically and via numerical simulations,
shows that whether prior commitment would be a viable evolutionary mechanism
for enhancing coordination and the overall population social welfare strongly
depends on the collective benefit and severity 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 group diversity is required for optimal coordination. The
results are robust for different selection intensities. Overall, our analysis
provides new insights into the complexity and beauty of behavioral evolution
driven by humans’ capacity for commitment, as well as for the design of self-
organised and distributed multi-agent systems for ensuring coordination among
autonomous agents.

Keywords: Commitment, Evolutionary Game Theory, Coordination, Technology
Adoption.

##  1 Introduction

Achieving a collective endeavour among individuals with their own personal
interest is an important social 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 and trust-based relationship among organisations and nations, it
success is often jeopardised by individual self-interest (Barrett et al.,,
2007; Perc et al.,, 2017). The study of mechanisms that support the evolution
of such collective behaviours has been of great interest in many disciplines,
ranging from Evolutionary Biology, Economics, Physics and Computer Science
(Nowak,, 2006; Sigmund,, 2010; Tuyls and Parsons,, 2007; West et al.,, 2007;
Han,, 2013; Perc et al.,, 2017; Andras et al.,, 2018; Kumar et al.,, 2020).
Several mechanisms responsible for the emergence and stability of collective
behaviours among such individuals, have been proposed, including kin and 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 inducing cooperative behaviour in the context
of pairwise and multi-player cooperation dilemmas (Nesse,, 2001; Frank,, 1988;
Han et al.,, 2017; Han et al., 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, ; Han et al.,, 2017; Kurzban et al.,, 2001). It provides an enhancement
to different forms of punishment against inappropriate behaviours and of
rewards to stimulate the appropriate ones (Chen et al.,, 2014; Martinez-
Vaquero et al.,, 2015, 2017; Sasaki et al.,, 2015; Powers et al.,, 2012;
Szolnoki and Perc,, 2012; Cimpeanu et al.,, 2019; Wang et al.,, 2019),
allowing ones to efficiently avoid free-riders (Han and Lenaerts,, 2016; Han
et al., 2015b, ) and resolve the antisocial punishment problem (Han,, 2016).
These works have primarily focused 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
desirable collective outcome to which all parties are required to commit if an
agreement is to be formed. The same argument is applied to other pairwise and
multi-player social dilemmas such as the Stag-Hunt and Chicken games, since
although the nature of the games is different from the PD and PGG, mutual
cooperation is the only desirable outcome to be achieved (Santos et al.,,
2006; Pacheco et al.,, 2009; Skyrms,, 2003). In other contexts such as
coordination problems, this is not the case anymore since there might be
multiple optimal or desirable collective outcomes and players might have
distinct, incompatible preferences regarding which outcome a mutual agreement
should aim to achieve (e.g. due to asymmetric benefits). Such coordination
problems are abundant in nature, ranging from collective hunting and foraging
to international climate change 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 mechanism for enhancing coordination and the population social welfare in
this type of coordination problems, in both 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 enabling an easier coordination of personal interests
(Nesse,, 2001; Cohen and Levesque,, 1990; Han,, 2013; Han et 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 technology investment strategic decision making problem,
allowing us to describe the models clearly. Namely, we describe technology
adoption games capturing the competitive market and decision-making process
among firms adopting new technologies (Zhu and Weyant,, 2003; Bardhan et al.,,
2004), with a key parameter $\alpha$ representing how competitive the market
is (thus describing how important coordination is). Similar to previous
commitment models, we will perform theoretical analysis and numerical
simulations resorting to stochastic methods from Evolutionary Game Theory
(EGT) (Hofbauer and Sigmund,, 1998; Sigmund et al.,, 2010).

We will start by modelling a pairwise technology adoption decision making,
where two investment firms (or players) competing within a same product market
who need to make strategic decision on which technology 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 firms invest on H they would end up competing with
each other leading to a smaller accumulated benefit than if they could
coordinate with each other to choose different technologies. However, given
the asymmetry in the benefits in such an outcome, clearly no firm would want
to commit to the outcome where its option is 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 interaction between more than two investment firms. In
the multi-player model, a key parameter $\mu$ is ascribed to the market demand
of high technology, i.e. what is the optimal fraction of the firms in a group
to adopt H. We analytically examine how players can be coordinated when there
is a market demand for a particular technology. We show that differently from
the two-player game, the newly defined parameter $\mu$ leads to a new kind of
complexity when trying to achieve group coordination. When there is a high
level of diversity in demand (i.e. intermediate values of $\mu$), as can be
seen in different technologies adoption contexts (Beede and Young,, 1998;
Schewe and Stuart,, 2015), introducing prior commitment can lead to
significant 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 details of the EGT methods for analysing them. Results of the
analysis and a final discussion will then follow.

##  2 Related Work

The problem of explaining the emergence and stability of collective behaviours
has been actively addressed in different disciplines (Nowak,, 2006; Sigmund,,
2010). Among other mechanisms, such as reciprocity and costly 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; Quillien,, 2020). It has shown that to enhance cooperation
commitments need to be sufficiently enforced and the cost of setting up the
commitments is justified with respect to the benefit derived 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; Arvanitis et 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
collective outcomes exist for which players have incompatible preferences; and
thus how benefits can be shared through compensations in order to resolve the
issues of asymmetric benefits, is crucially important (Bianca and 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 investigated how coordination and cooperation
can be improved using prior commitment deal when there are multiple players
involved and also when there is a particular market demand (Bianca and Han,,
2019). Our approach in exploring how implementing prior commitment enhances
cooperation dilemma has also been investigated by previous researchers in the
past (Chen and Komorita,, 1994). A good level of cooperation was seen in a
Public Good Game experiment when there was a binding agreement made during the
prior communication stage among members of the group. They hypothesized that
if members of a group are allowed to make a pledge (a degree of
bindings/commitment) before their actual decisions, they will be able to
communicate their intentions and it will overall increase cooperation rate in
the population. As predicted, their results clearly demonstrate that making a
pledge improves cooperation although the degree of commitment required in the
pledge deferentially affected the cooperation rate (Chen and Komorita,, 1994;
Cherry and McEvoy,, 2013; Kurzban et al.,, 2001).

There have been several other works studying the evolution of coordination,
using the so-called Stag Hunt game, see e.g. (Skyrms,, 2003; Pacheco et al.,,
2009; Santos et al.,, 2006; Sigmund,, 2010). However, to the best of our
knowledge there has been no work studying how prior commitments can be
modelled and used for enhancing the outcome of the evolution of coordination.
As our results below show, significant enhancement of coordination and
population welfare can be achieved via the arrangement of suitable commitment
deals.

Furthermore, it is noteworthy that commitments have been studied extensively
in Artificial Intelligence and Multi-agent systems literature, see e.g.
(Singh,, 1991; Castelfranchi and Falcone,, 2010; Chopra and Singh,, 2009;
Rzadca et al.,, 2015; Harrenstein et al.,, 2007; Winikoff,, 2007). Differently
from our work, these studies utilise commitments for the purpose of regulating
individual and collective behaviours, formalising different aspects of
commitments (such as norms and conventions) in multi-agent systems. However,
our results and approach provide important new insights into the design of
such systems as these require commitments to ensure high levels of efficient
collaboration and coordination within a group or team of agents. For example,
by providing suitable agreement deals agents can improve the chance that a
desirable collective outcome (which is best for the systems as a whole) is
reached even when benefits provided by the outcome are different for the
parties involved.

##  3 Models and Methods

In the following , we first describe a two-player technology adoption game
then extend it with the option of arranging 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
Evolutionary Game Theory for finite populations, which will be used to analyse
the resulting models.

###  3.1 Two-player Tech Adoption (TD) Game

####  3.1.1 Two-player TD Without Commitments

We consider the scenario that two firms (players) compete for the same product
market, and they need to 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 investments
by the following payoff matrix (for row player):

$\bordermatrix{~{}&H&L\cr H&\alpha b_{H}-c_{H}&b_{H}-c_{H}\cr
L&b_{L}-c_{L}&\alpha b_{L}-c_{L}\cr}=\bordermatrix{~{}&H&L\cr H&a&b\cr
L&c&d\cr},$ (1)

where $c_{L}$, $c_{H}$ and $b_{L}$, $b_{H}$ ($b_{L}\leq b_{H}$) represent the
costs and benefits of investing on L and H, respectively; $\alpha\in(0,1)$
indicates the competitive level of the product market: the firms receive a
partial benefit if they both choose to invest on the same technology.
Collectively, the smaller $\alpha$ is (i.e. the higher the market
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 generally applicable to many other coordination
problems for instance wherever there are strategic investment decisions to
make (in competitive markets of any products) (Zhu and Weyant,, 2003;
Chevalier-Roignant et al.,, 2011).

####  3.1.2 Two-player TD in Presence of Commitments

We now extend the model allowing players to have the option to arrange a prior
commitment before a TD 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 these
commitment proposing strategies as HP and LP, respectively. Similarly to
previous models of commitments (for PD and PGG) (Han et al.,, 2013; Han et
al., 2015a, ), to make the commitment deal reliable, a proposer pays an
arrangement cost $\epsilon$. If the co-player agrees with the deal, then the
proposer assumes that the opponent will adopt the agreed choice, yet there is
no guarantee that this will actually be the case. Thus whenever a co-player
refuses to commit, HP and LP would play H in the game. When the co-player
accepts the commitment though later does not honour it, she has to compensate
the honouring co-player at a personal cost $\delta$.

Differently from previous models on PD and PGG where an agreed outcome leads
to the same payoff for all parties in the agreement (mutual cooperation
benefit), in the current model such an outcome would lead to different payoffs
for those involved. Therefore, as part of the agreement, HP would compensate
after the game an amount $\theta_{1}$ to accepted player that honours the
agreement; while LP would request a compensation $\theta_{2}$ from such an
accepted co-player.

Besides HP and LP, we consider a minimal model with the following (basic)
strategies in this commitment version:

  * •

Non-proposing acceptors, HC and LC, who always commit when being proposed a
commitment deal wherein they are willing to adopt any technology proposed
(even when it is different from their intended 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 the game, and do not propose commitments;

  * •

Fake committers, HF and LF, who accept a commitment proposal yet play the
choice opposite to what has been agreed whenever the game takes place. These
players assume that they can exploit the commitment proposing players without
suffering the consequences 111Compared to cooperation dilemmas such as PD and
PGG, fake strategies make less sense in the context of coordination games
since they would not earn the temptation payoff by adopting a different choice
from what being agreed. Moreover, in the 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 them from multi-player
settings for simplicity, without being detrimental to the results. .

Note that similar to the commitment models for the PD game (Han et al.,,
2013), some possible strategies have been excluded from the analysis since
they are dominated by at least one of the strategies in any configuration of
the game: they can be omitted without changing the outcome of the analysis.
For example, those who propose a commitment (i.e. paying a cost $\epsilon$)
but then do not honour (thus have to pay the compensation when facing a
honouring acceptors) would be dominated by the corresponding non-proposers.

Together the model consists of eight strategies that define the following
payoff matrix, capturing the average payoffs that each strategy will receive
upon interaction with one of the other seven strategies (where we denote
$\lambda=\theta_{1}+\theta_{2}$, $\lambda_{1}=b-\epsilon-\theta_{1}$,
$\lambda_{2}=c-\epsilon+\theta_{2}$, $\lambda_{3}=a-\epsilon+\delta$ and
$\lambda_{4}=d-\epsilon+\delta$, just for the sake of clear representation)

$\bordermatrix{~{}&\text{HP}&\text{LP}&\text{HN}&\text{LN}&\text{HC}&\text{LC}&\text{HF}&\text{LF}\cr\text{HP}&\frac{b+c-\epsilon}{2}&\frac{2b-\epsilon-\lambda}{2}&a&b&\lambda_{1}&\lambda_{1}&\lambda_{3}&\lambda_{3}\cr\text{LP}&\frac{2c-\epsilon+\lambda}{2}&\frac{b+c-\epsilon}{2}&a&b&\lambda_{2}&\lambda_{2}&\lambda_{4}&\lambda_{4}\cr\text{HN}&a&a&a&b&a&b&a&b\cr\text{LN}&c&c&c&d&c&d&c&d\cr\text{HC}&c+\theta_{1}&b-\theta_{2}&a&b&a&b&a&b\cr\text{LC}&c+\theta_{1}&b-\theta_{2}&c&d&c&d&c&d\cr\text{HF}&a-\delta&d-\delta&a&b&a&b&a&b\cr\text{LF}&a-\delta&d-\delta&c&d&c&d&c&d\cr}.$
(2)

Note that when two commitment proposers interact only one of them will need to
pay the cost of setting up the commitment. Yet, as either one of them can take
this action they pay this cost only half of the time (on average). In
addition, the average payoff of HP when interacting with LP is given by
$\frac{1}{2}(b-\epsilon-\theta_{1}+b-\theta_{2})=\frac{1}{2}(2b-\epsilon-\theta_{1}-\theta_{2})$.
When two HP players interact, each receives
$\frac{1}{2}(b-\epsilon-\theta_{1}+c+\theta_{1})=\frac{1}{2}(b+c-\epsilon)$.

We say that an agreement is fair if both parties obtain the same benefit when
they honour it (after having taken into account the cost of setting up the
agreement). For that, we can show that $\theta_{1}$ and $\theta_{2}$ must
satisfy $\theta_{1}=\frac{b-c-\epsilon}{2}$ and
$\theta_{2}=\frac{b-c+\epsilon}{2}$, and thus, both parties obtain
$\frac{b+c-\epsilon}{2}$. Indeed, they can be achieved by comparing the
payoffs of HP and HC when they interact, i.e.
$b-{\epsilon}-\theta_{1}=c+\theta_{1}$, where solving this equation we would
obtain $\theta_{1}=\frac{b-c-\epsilon}{2}$.

With these conditions it also ensures that the payoffs of HP and LP when
interacting with each other are equal. Our analysis below will first focus on
whether and when the fair agreements can lead to improvement in terms of
coordination and the overall social welfare (i.e. average population payoff).
We will discuss how different kinds of agreements (varying $\theta_{1}$ and
$\theta_{2}$) affect the outcome, with additional results provided in
Appendix.

###  3.2 Multi-Player Tech Adoption (TD) Game

####  3.2.1 Multi-player TD Without Commitments

We now describe a $N$-player ($N>2$) version of the TD model. Again, as
before, we will introduce the model 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 written as follows

$\begin{split}\Pi_{H}(k)&=\alpha_{H}(k)b_{H}-c_{H},\\\
\Pi_{L}(k)&=\alpha_{L}(k)b_{L}-c_{L},\end{split}$ (3)

where $\alpha_{H}(k)$ and $\alpha_{L}(k)$ 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 $\alpha$.
To generalize for $N$-player TD interactions, they should also depend on the
demand for high technology (H) in the group, describing what is the maximal
number of players in the group that can adopt H without reducing their benefit
due to competition. Let us denote this number by $\mu$ (where $1\leq\mu\leq
N$). For example, intermediate values of $\mu$ indicate a high level of group
diversity is needed for optimal coordination. When $\mu=N$, it means there is
a significant market demand of the high benefit technology so that all firms
can adopt it without leading to competition.

Hence, we define

$\alpha_{H}(k)=\begin{cases}1,&\text{if $k\leq\mu$},\\\
\frac{\alpha_{1}\mu}{k}&\text{otherwise},\end{cases}$ (4)

$\alpha_{L}(k)=\begin{cases}1,&\text{if $k\geq\mu$},\\\
\frac{\alpha_{2}(N-\mu)}{N-k}&\text{otherwise}.\end{cases}$ (5)

The rationale of these definitions is that whenever $k\leq\mu$, full benefits
from adopting H can be obtained, and moreover, if $k>\mu$, the larger $k$ the
stronger the competition is among H adopters. Similarly for L adopters. The
parameters $\alpha_{1}$ and $\alpha_{2}$ stand for the intensities of
competition for investing in H and in L, respectively. For simplicity we
assume in this paper $\alpha_{1}=\alpha_{2}=\alpha$. Note that for $N=2$ we
recover the two-player model given in Equation (1) , given that the current
$\alpha$ is scaled (by 2) compared to the value of $\alpha$ in the pairwise
game, solely for the purpose of a clear presentation.

The optimal group payoff is achieved when there are exactly $\mu$ players
adopting H and the rest adopting L, leading to an average payoff for each
member given by

$A:=\frac{\mu(b_{H}-c_{H})+(N-\mu)(b_{L}-c_{L})}{N}.$

####  3.2.2 Multi-player TD in Presence of Commitments

We can define the $N$-player game version with prior commitments in a similar
fashion as in the two-player game. Commitment proposing strategists (i.e. HP
and LP players) will propose before an interaction that the group will play
the optimal arrangement (so that every player obtains an average payoff $A$).
For simplicity, we assume that the committed players adopt the fair agreement,
i.e. every member will obtain the same payoff after compensation is made to
those adopting L. As such, we don’t need to consider who will adopt H or L, as
all would receive the same payoff at the end. Moreover, whenever a player in
the group refuses to commit, commitment proposers will adopt H. Details of
payoff calculation will be provided in Results section (cf. Table 1).

###  3.3  Evolutionary Dynamics

In this work, we will perform theoretical analysis and numerical simulations
(see next section) using EGT methods for finite populations (Nowak et al.,,
2004; Imhof et al.,, 2005; Hauert et al.,, 2007). Let $Z$ be the size of the
population. In such a setting, individuals’ payoff represents their _fitness_
or social _success_ , and evolutionary dynamics is shaped by social learning
(Hofbauer and Sigmund,, 1998; Sigmund,, 2010), whereby the most successful
individuals will tend to be imitated more often by the other individuals. In
the current work, social learning is modelled using the so-called pairwise
comparison rule (Traulsen et al.,, 2006), a standard approach in EGT, assuming
that an individual $A$ with fitness $f_{A}$ adopts the strategy of another
individual $B$ with fitness $f_{B}$ with probability $p$ given by the Fermi
function,

$p_{A,B}=\left(1+e^{-\beta(f_{B}-f_{A})}\right)^{-1}.$

The parameter $\beta$ represents the ‘imitation strength’ or ‘intensity of
selection’, i.e., how strongly the individuals base their decision to imitate
on fitness difference between themselves and the opponents. For $\beta=0$, we
obtain the limit of neutral drift – the imitation decision is random. For
large $\beta$, imitation becomes 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, with a certain mutation
probability, an individual switches randomly to a different strategy without
imitating another individual. In the limit of small mutation rates, the
dynamics will proceed with, at most, two strategies in the population, such
that the behavioural dynamics can be conveniently described by a Markov Chain,
where each state represents a monomorphic population, whereas the transition
probabilities are given by the fixation probability of a single mutant (Imhof
et al.,, 2005; Nowak et al.,, 2004; Hauert et al.,, 2007). The resulting
Markov Chain has a stationary distribution, which characterises the average
time the population spends in each of these monomorphic end states. It has
been shown to have a range of applicability which goes well beyond the strict
limit of very small mutation (or exploration) rates (Hauert et al.,, 2007;
Sigmund,, 2010; Han et al.,, 2012; Sigmund et al.,, 2010; Rand et al.,, 2013).

Before describing how to calculate this stationary distribution, we need to
show how payoffs are calculated, which differ for two-player and N-player
settings, as below.

  * $\bullet$

Average Payoff for the Two Player Game
Let $\pi_{ij}$ 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 strategies in the
population, say, ${x}$ individuals using $i$ ($0\leq{x}\leq Z$) and ($Z-{x}$)
individuals using $j$. Thus the average payoff of the individual that uses $i$
or $j$ can be written respectively as follows

$\begin{split}\Pi_{i}({x})&=\frac{({x}-1)\pi_{ii}+(Z-{x})\pi_{i,j}}{Z-1},\\\
\Pi_{j}({x})&=\frac{{x}\pi_{j,i}+(Z-{x}-1)\pi_{j,j}}{Z-1}.\end{split}$ (6)

  * $\bullet$

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 of type $j$, in $N$
trails, is given by the hypergeometric distribution as follows (Sigmund,,
2010; Gokhale and Traulsen,, 2010)

$H(k,N,x,{Z})=\frac{\binom{x}{k}\binom{Z-x}{N-k}}{\binom{Z}{N}}$

Hence, in a population of $x$ $i$-strategists and $(Z-x)$ $j$ strategists, the
average payoff of $i$ and $j$ are given by

$\begin{split}\Pi_{ij}({x})&=\sum_{k=0}^{N-1}H(k,N-1,x-1,Z-1)\pi_{ij}(k+1)=\sum_{k=0}^{N-1}\frac{\binom{x-1}{k}\binom{Z-x}{N-1-k}}{\binom{Z-1}{N-1}}\pi_{ij}(k+1),\\\
\Pi_{ji}({x})&=\sum_{k=0}^{N-1}H(k,N-1,x-,Z-1)\pi_{ji}(k)=\sum_{k=0}^{N-1}\frac{\binom{x}{k}\binom{Z-1-x}{N-1-k}}{\binom{Z-1}{N-1}}\pi_{{ji}}(k).\end{split}$
(7)

Now, for both two-player and $N$-player settings, the probability to change
the number ${x}$ of individuals using strategy A by $\pm$ one in each time
step can be written as (Traulsen et al.,, 2006)

$T^{\pm}(k)=\frac{Z-{x}}{Z}\frac{{x}}{Z}\left[1+e^{\mp\beta[\Pi_{i}({x})-\Pi_{j}({x})]}\right]^{-1}.$
(8)

The fixation probability of a single mutant with a strategy $i$ in a
population of $({Z}-1)$ individuals using $j$ is given by (Traulsen et al.,,
2006; Nowak et al.,, 2004)

$\rho_{j,i}=\left(1+\sum_{i=1}^{{Z}-1}\prod_{j=1}^{i}\frac{T^{-}(j)}{T^{+}(j)}\right)^{-1}.$
(9)

Considering a set $\\{1,...,q\\}$ of different strategies, these fixation
probabilities determine a transition matrix $M=\\{T_{ij}\\}_{i,j=1}^{q}$, with
$T_{ij,j\neq i}=\rho_{ji}/(q-1)$ and $T_{ii}=1-\sum^{q}_{j=1,j\neq i}T_{ij}$,
of a Markov Chain. The normalised eigenvector associated with the eigenvalue 1
of the transposed of $M$ provides the stationary distribution described above
(Imhof et al.,, 2005), describing the relative time the population spends
adopting each of the strategies.

##### Risk-dominance

An important measure to determine the evolutionary dynamic of a given strategy
is its risk-dominance against others. For the two strategies $i$ and $j$,
risk-dominance is a criterion which determine which selection direction is
more probable: an $i$ mutant is able to fixating in a homogeneous population
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 limit for large population size $Z$ when

$\sum_{k=1}^{N}\Pi_{i,j}(k)\geq\sum_{k=0}^{N-1}\Pi_{j,i}(k)$ (10)

This condition is applicable for both two-player games, $N=2$, and when
N-player games with $N>2$ (Sigmund,, 2010; Gokhale and Traulsen,, 2010). It
will allow us to derive analytical conditions such as when commitment
proposing is an evolutionarily viable strategy, being risk-dominant against
all other strategies in the population.

Parameters description | Notation
---|---
Cost of investing in high technology, H | $c_{H}$
Cost of investing in low technology, L | $c_{L}$
Benefit of investing in high technology, H | $b_{H}$
Benefit of investing in low technology, L | $b_{L}$
Competitive level of the market | $\alpha$
Group size (in N-player TD games) | $N$
Optimal number of H-adopters in a group of N players | $\mu$
Cost of arranging a commitment | $\epsilon$
Compensation paid by dishonouring commitment acceptors | $\delta$
Compensation paid by HP to honouring commitment acceptors | $\theta_{1}$
Compensation paid to LP by commitment acceptors | $\theta_{2}$
Table 1:  List of parameters in the models.

##  4 Results

We will first describe results for two-player games, then proceeding to
provide those for the $N$-player version. Table 1 summarizes the key
parameters in both versions, for ease of following.

###  4.1 Two-player TD Game Results

####  4.1.1 Analytical Conditions for the Viability of Commitment Proposers

To begin with, using the conditions given in Equation 10, we obtain that if

$\theta_{1}+\theta_{2}<b-c$

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 are evolutionarily 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 and
only if

$\begin{split}\epsilon&<\min\\{b+c-2a,3b-c-2d,\frac{3b-c-2a-4\theta_{1}}{3},\frac{3b-c-2d-4\theta_{1}}{3},\frac{b+c-2a+4\delta}{3},\frac{b+c-2d+4\delta}{3}\\},\\\
\epsilon&<\min\\{b+c-2a,3b-c-2d,\frac{3c-b-2a+4\theta_{2}}{3},\frac{3c-b-2d+4\theta_{2}}{3},\frac{b+c-2a+4\delta}{3},\frac{b+c-2d+4\delta}{3}\\}.\end{split}$
(11)

Note that each element in the min expressions above corresponds to the
condition for one of the six non-proposing strategies HN, LN, HC, LC, HF, LF,
respectively.

Thus, we can derive the conditions for $\theta_{1}$, $\theta_{2}$ and
$\delta$:

$\begin{split}\theta_{1}&<\frac{1}{4}\left(3b-c-3\epsilon-2\max\\{a,d\\}\right),\\\
\theta_{2}&>\frac{1}{4}\left(b-3c+3\epsilon+2\max\\{a,d\\}\right),\\\
\delta&>\frac{1}{4}\left(3\epsilon-b-c+2\max\\{a,d\\}\right).\end{split}$ (12)

In particular, for fair agreements, i.e. $\theta_{1}=(b-c-\epsilon)/2$ and
$\theta_{2}=(b-c+\epsilon)/2$, we obtain

$\begin{split}&\epsilon<b+c-2\max\\{a,d\\},\\\
\delta&>\frac{1}{4}\left(3\epsilon-b-c+2\max\\{a,d\\}\right).\end{split}$ (13)

It is because $3b-c-2d>b+c-2\max\\{a,d\\}$, which is due to $b>c$ and
$\max\\{a,d\\}\geq d$.

In general, these conditions indicate that for commitments to be a viable
option for improving coordination, the cost of arrangement $\epsilon$ must be
sufficiently small while the compensation associated with the contract needs
to be sufficiently large (see already Figure 2 for numerical validation).
Furthermore, for the first condition to hold, it is necessary that
$b+c>2\max\\{a,d\\}$. It means that the total payoff of two players when
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 $\alpha$
and the costs and benefits of investment, as follows (see again the payoff
matrices in Equation 1)

$\begin{split}&\alpha<{\frac{1}{2}}+\min\\{\frac{c_{H}+b_{L}-c_{L}-\epsilon}{2b_{H}},\frac{c_{L}+b_{H}-c_{H}-\epsilon}{2b_{L}}\\},\\\
&\alpha<{\frac{1}{2}}+\min\\{\frac{c_{H}+b_{L}-c_{L}-3\epsilon+4\delta}{2b_{H}},\frac{c_{L}+b_{H}-c_{H}-3\epsilon+4\delta}{2b_{L}}\\},\end{split}$

which can be rewritten as

$\begin{split}&\alpha<{\frac{1}{2}}+\min\\{\frac{c_{H}+b_{L}-c_{L}-\max\\{\epsilon,3\epsilon-4\delta\\}}{2b_{H}},\frac{c_{L}+b_{H}-c_{H}-\max\\{\epsilon,3\epsilon-4\delta\\}}{2b_{L}}\\}.\end{split}$
(14)

This condition indicates under what condition of the market competitiveness
and the costs and benefits of investing 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, $\epsilon$, leads to a
smaller threshold of $\alpha$ where commitment is viable. Moreover, given a
commitment system (i.e. fixing $\epsilon$ and $\delta$), assuming similar
costs of investment for the two technologies, then a larger ratio of the
benefits obtained from the two technologies, $b_{H}/b_{L}$, leads to a smaller
upper bound for $\alpha$ for which commitment is viable.

Remarkably, our numerical analysis below (see already Figure 1) shows that the
condition in Equation 14 accurately predicts the threshold of $\alpha$ where
commitment proposing strategies (i.e. HP and LP) are highly abundant in the
population, leading to improvement in terms of the average population payoff
compared to when commitment is absent (Figure 3). For example, when
$\epsilon=0.1,\ 1$ and $2$, the upper bounds for $\alpha$ are $0.658$, $0.583$
and $0.5$, respectively.

On the other hand, when $\alpha$ 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).

Figure 1: Frequency of the eight strategies, HP, LP, HN, LN, HC, LC, HF and
LF, as a function of $\alpha$, for different values of $\epsilon$ and $\beta$.
In general, the commitment proposing strategies HP and LP dominate the
population when $\alpha$ is small while HN and HC dominate when $\alpha$ is
sufficiently large in all cases, which is robust for different values of
intensity of selection, $\beta$. The HN and HC dominate the population as the
market competition decreases (i.e. when $\alpha$ increases). Larger values of
$\beta$ increase the difference between strategies’ frequencies but do not
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:
$\delta=6$; $\beta=0.01,0.1$ and 1; population size $Z=100$; Fair agreements
are used, where $\theta_{1}$ and $\theta_{2}$ are given by
$\theta_{1}=(b-c-\epsilon)/2$ and $\theta_{2}=(b-c+\epsilon)/2$.  Figure 2:
Total frequency of commitment strategies (i.e. sum of the frequencies of HP
and LP), as a function of $\epsilon$ and $\delta$, for different values of
$\alpha$ and $\beta$. Primarily, the commitment proposing strategies dominate
the population whenever $\epsilon$ is sufficiently small and $\delta$ is
sufficiently large. Furthermore, the smaller $\alpha$, these commitment
strategies dominate for a wider range of $\epsilon$ and $\delta$, especially
when $\alpha$ is smaller. These observations are robust for different values
of $\beta$. Nevertheless, a larger $\beta$ leads to a greater frequency of
commitment proposing strategies where they are evolutionarily viable, and a
lower frequency otherwise. 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:
$\beta=0.01$ in the first, $\beta=0.1$ in the second and $\beta=1$ in the
third row; population size $Z=100$; Fair agreements are used, where
$\theta_{1}$ and $\theta_{2}$ are given by $\theta_{1}=(b-c-\epsilon)/2$ and
$\theta_{2}=(b-c+\epsilon)/2$.  Figure 3: Average population payoff as a
function of $\alpha$, when commitment is absent and when it is present, for
different values of $\epsilon$ and $\beta$. We observed that when $\alpha$ is
small, significant improvement in terms of the average population payoff can
be achieved through prior commitment. When $\alpha$ is sufficiently large,
commitment leads to no improvement or might even be detrimental for social
welfare, especially when $\beta$ is small. That is, at $\alpha=0.7$ in panel a
and $\alpha=0.9$ in panel d, without commitment will be more beneficial.
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 $\beta=0.01,\ 0.1$ and $1$
respectively. Also, in panels d, e and f) $b_{H}=3$ (i.e. $b=2$) with
$\beta=0.01,\ 0.1$ and $1$ respectively; Other parameters: $\delta=6$;
population size $Z=100$; Fair agreements are used, where $\theta_{1}$ and
$\theta_{2}$ are given by $\theta_{1}=(b-c-\epsilon)/2$ and
$\theta_{2}=(b-c+\epsilon)/2$.

####  4.1.2 Numerical Results for Pairwise TD game

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 a function of $\alpha$, for
different values of $\epsilon$ and game configurations. In general, the
commitment proposing strategies HP and LP dominate the population when
$\alpha$ is small while HN and HC dominate when $\alpha$ is sufficiently large
even with different values of $\beta$ utilized in the comparison. That is,
commitment proposing strategies are viable and successful whenever the market
competitiveness is high, leading to the need of efficient coordination among
the competing players/firms to ensure high benefits. Notably, we observe that
the thresholds of $\alpha$ below which HP and LP are dominant, closely
corroborate the analytical condition described in Equation 14, in all cases.
This observation is also robust for different values of intensity of
selection, $\beta$.

This observation is robust for varying commitment parameters, i.e. the cost of
arranging commitment, $\epsilon$, and the compensation cost associated with
commitment, $\delta$, see Figure 2. Namely, we show the total frequency of
commitment strategies (i.e. sum of the frequencies of HP and LP) for varying
these parameters and for different values of $\alpha$. It can be seen that, in
general, the commitment strategies dominate the population whenever $\epsilon$
is sufficiently small and $\delta$ is sufficiently large. This observation is
in accordance with previous commitment modelling works for the cooperation
dilemma games (Han et al.,, 2013; Han et al., 2015a, ; Han et al.,, 2017). In
addition, we observe that in the current coordination problem, that the
smaller $\alpha$ is, these commitment strategies dominate the population for
wider range of $\epsilon$ and $\delta$. Our additional results show that these
observations are robust with respect to other game configurations, including
$\beta$ (comparing the three rows in Figure 2.

Now, in order to determine whether and when commitments can actually lead to
meaningful improvement, in Figure 3, we compare the average population payoff
or social welfare when a commitment is present and when it is absent. In
general, it can be seen that when $\alpha$ is sufficiently small (below a
threshold), the smaller it is, the greater improvement of social welfare is
achieved through the presence of a commitment deal. Moreover, the smaller the
cost of arranging commitments, $\epsilon$, the greater improvement is
obtained. When $\alpha$ is sufficiently large, commitment leads to no
improvement or might even be detrimental for social welfare, especially when
$b_{H}/b_{L}$ is large (which is in accordance with the analytical results
above). The detriment is further increased when $\beta$ is small. We can
observe that the thresholds for which a notable improvement can be achieved is
the same as the one for the viability of HP and LP (i.e. as described in
Equation 14).

###  4.2 Multiplayer Game Results

####  4.2.1 Payoff Derivation in $N$-player TD game

As mentioned above, compared to cooperation dilemmas such as PD and PGG, fake
strategies make less sense in the context of coordination games since they
would not earn the temptation payoff by adopting a different choice from what
being agreed. To focus on the group effect and the effect of the newly
introduced parameter $\mu$, we will consider a population consisting of HP,
LP, HN, LN, HC and LC (i.e. excluding fake strategies). As shown in the two-
player game analysis, the fake strategies (i.e. HF and LF) are not viable
options in TD games and can be ignored. It is equivalent to consider to the
full set of strategies with a sufficiently large $\delta$.

First of all, we derive the payoffs received by each strategy when
encountering specific other strategies (see a summary in Table 2). Namely,
$\Pi_{ij}(k)$ and $\Pi_{ji}(k)$ 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 column of the table lists all possible
strategies which can be used by player $i$ (focal player), where as the second
column shows strategies of co-players (opponents). The third column shows the
payoffs of focal players.

Focal Player ($i$) | Opponent ($j$) | $\Pi_{i,j}(k)$
---|---|---
HP, LP | HP, LP | $A-\epsilon/N$
HP, LP | HC, LC | $A-\epsilon/k$
HP, LP | HN |  $\Pi_{H}(N)$ (for $k<N$)
HP, LP | LN |  $\Pi_{H}(k)$ (for $k<N$)
HN | HP, LP, HN, HC | $\Pi_{H}(N)$
HN | LN, LC | $\Pi_{H}(k)$
LN | HP, HN, HC | $\Pi_{L}(k)$
LN | LN, LC | $\Pi_{L}(N)$
LN | LP | $\Pi_{L}(k)$
HC, LC | HP, LP | A (for $k<N$)
HC | HN, HC | $\Pi_{H}(N)$
HC | LN, LC | $\Pi_{H}(k)$
LC | HN, HC | $\Pi_{L}(k)$
LC | LN, LC | $\Pi_{L}(N)$
Table 2:  Average payoffs of focal strategy $i$ when facing strategy $j$, in a
group of $k$ former and $N-k$ latter strategists.

####  4.2.2 Analytical conditions for the viability of commitment proposers in
N-player TD game

We now derive the conditions under which HP is risk-dominant against the rest
of strategies. Since we assume fair agreements, the conditions for LP would be
equivalent to those for HP in terms of risk-dominance. For ease of following
the derivations below, we recall that $A$ denotes the optimal group payoff
achieved when there are exactly $\mu$ players adopting H and the rest adopting
L, that is,
$A:=\frac{1}{N}\left(\mu(b_{H}-c_{H})+(N-\mu)(b_{L}-c_{L})\right)$.

HP is risk-dominant against HC if

$\sum_{k=1}^{N}\Pi_{HP,HC}(k)\geq\sum_{k=0}^{N-1}\Pi_{HC,HP}(k),$

which can be written as

$\sum_{k=1}^{N}\left(A-\frac{\epsilon}{k}\right)\geq\Pi_{H}(N)+\sum_{k=1}^{N-1}A.$

Hence we obtain

$\epsilon\leq\frac{A-\Pi_{H}(N)}{H_{N}},$ (15)

where $H_{N}=\sum_{k=1}^{N}\frac{1}{k}$.

Figure 4: Frequency of the six strategies HP, LP, HN, LN, HC and LC, as a
function of $\epsilon$ in a N-player game with commitment, for different
values of $\mu$. In the N-player game, the new parameter $\mu$ describes the
market demand for a high technology, which was set to 1 in the pairwise game.
HP and LP have a high frequency for sufficiently small $\epsilon$ for $\mu=2$
in both games, and also when $\mu=1$ for the first, easy coordinate situation
(first row). When $\mu=5$, i.e. when all players can adopt H without benefit
reduction, HC always dominate and commitment strategies are not successful.
This means that when there is a need for a diversity of technology adoption,
initiating prior commitments to enhance coordination is important. Parameters:
in panel a, b and c) $b_{H}=6$ (i.e. $b=5$) with $\mu=1,2,5$ respectively.
Also, in panel d,e and f) $b_{H}=3$ (i.e. $b=2$) with $\mu=1,2,5$
respectively; Other parameters: $N=5$, $\beta=0.1$; $\alpha=0.5$; $c_{H}=1$,
$c_{L}=1$, $b_{L}=2$ (i.e. $c=1$); .

Similarly, HP is risk-dominant against LC if

$\epsilon\leq\frac{A-\Pi_{L}({0})}{H_{N}}.$ (16)

For risk-dominance of HP against HN,

$\sum_{k=1}^{N}\Pi_{HP,HN}(k)\geq\sum_{k=0}^{N-1}\Pi_{HN,HP}(k),$

which equivalently can be written as

$A-\frac{\epsilon}{N}\geq\Pi_{H}(N),$

or,

$\epsilon\leq N\big{(}A-\Pi_{H}(N)\big{)}.$ (17)

Finally, HP is risk-dominant against LN if

$\sum_{k=1}^{N}\Pi_{HP,LN}(k)\geq\sum_{k=0}^{N-1}\Pi_{LN,HP}(k),$

which can be rewritten as

$A-\frac{\epsilon}{N}+\sum_{k=1}^{N-1}\Pi_{H}(k)\geq\sum_{k=0}^{N-1}\Pi_{L}(k),$

or

$\epsilon\leq
N\left(A+\sum_{k=1}^{N-1}\Pi_{H}(k)-\sum_{k=0}^{N-1}\Pi_{L}(k)\right).$ (18)

In short, in order for commitment proposers to be risk-dominant against all
other strategies, it requires that $\epsilon$ is sufficiently small, namely,
smaller than minimum of the right hand sides of Equations (15)-(18).

Figure 5: Frequency of the six strategies HP, LP, HN, LN, HC and LC, as a
function of $\alpha$ in a multiplayer game with commitment, for different
values of $\epsilon$ and also two different game configurations. In general,
the commitment proposing strategies (HP and LP) decrease in frequency for
increasing $\alpha$. They dominate over other strategies for sufficiently
small $\alpha$ and $\epsilon$. That is, it is more beneficial to engage in a
prior commitment deal 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 $\epsilon=0.1,\ 1$ and 2, respectively. Also, in panel d, e and f:
$b_{H}=3$ (i.e. $b=2$) with $\epsilon=0.1,1$ and $2$ respectively; Other
parameters: $N=5$, $\beta=0.1$; $\mu=2$. Figure 6: Total frequency of
commitment proposing strategies HP and LP as a function of $\mu$ and
$\epsilon$. In general, the commitment proposing strategies are most
successful for intermediate values of $\mu$, especially for a sufficiently
small cost of arranging prior commitment $\epsilon$. 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: $N=5$,
$\beta=0.1$; $\alpha=0.5$. Figure 7: Average population payoff (social
welfare) as a function of $\mu$ with different values of $\epsilon$, showing
when commitment is absent against when it is present. We compare results for
different values of $\beta$ in two game configurations. We observe that
whenever $\mu<5$ (i.e. when there is a need for coordination to avoid
competition in the group), arranging a prior commitment is beneficial to the
population social welfare. Parameters: in panel a, b and c) $b_{H}=6$ (i.e.
$b=5$), in panel d,e and f) $b_{H}=3$ (i.e. $b=2$). Other parameters: $N=5$,
$\alpha=0.5$, $c_{H}=1$, $c_{L}=1$, $b_{L}=2$

###  4.3 Numerical Results for N-player TD game

We compute stationary distributions in a population of six strategies HP, LP,
HN, LN, HC and LC, for the 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 conditions
obtained in the previous section regarding when commitment proposing
strategies are evolutionarily viable strategies (being risk-dominant against
others). Similar to the pairwise TD game, we observe that there is a threshold
for $\epsilon$ below which it is the case. Moreover, Figure 5 shows that the
frequencies of these strategies (HP and LP) decrease for increasing $\alpha$.
They dominate the population whenever $\epsilon$ is sufficiently small (e.g.
$\epsilon=0.1$ and $1$). That is, it is more beneficial to engage in a prior
commitment deal when the market competition is harsher (i.e. small $\alpha$).
These results are robust for different intensities of selection (see Figure 10
in Appendix). In general, our results confirm the similar observations
regarding the effects of $\epsilon$ and $\alpha$ on the evolutionary outcomes
obtained in the pairwise game above.

We now focus on understanding the effect of the new parameter in the N-player
game, $\mu$, on the evolutionary outcomes. Recall that $\mu$ indicates the
demand for high technology (H) in the group, describing what is the maximal
number of players in the group that can adopt H without reducing their benefit
due to competition. Figure 4 shows the effect of different values of $\mu$ on
the frequency or evolutionary success of all strategies as a function of
$\epsilon$. When $\mu$ is small to intermediate, and the cost of arranging
prior commitment is also small, the commitment proposing strategies are
dominant. This suggests that arranging prior commitments might be more
beneficial in such instances. These results also imply that $\mu$ is very
essential in determining when commitment should be initiated. Apparently, the
greater need for a group mixture or market diversity of technologies,
indicating a more difficult coordination situation, the greater need for the
utilization of commitment to enhance coordination among players is. This
observation is even more evident in Figure 6, where we examine the success of
commitment for varying $\mu$ and $\epsilon$, in regards to two different game
configurations. It can be observed that an intermediate value of $\mu$ leads
to the highest frequency of commitment strategies, especially in the more
difficult coordination situation (i.e. the right panel).

We now closely examine the gain in terms of social welfare improvement when
using prior commitments. As shown in Figure 7, whenever $\mu<N$ ($N=5$), i.e.
there is a need to coordinate among the group players to avoid competition
that induces benefit reduction, prior commitments lead to increase of social
welfare. This increase is more significant in the more difficult coordination
situation (i.e. the lower row) and when the cost of arranging commitment is
low, which is also slightly more significant for intermediate values of $\mu$
and higher values of intensity of selection, $\beta$.

##  5  Conclusions and Further Discussion

We have described in this paper novel evolutionary game theory models showing
how prior commitments can be adopted as an efficient mechanism for enhancing
coordination, in both pairwise and multi-player interactions. For that, we
described technology adoption (TD) games where technology investment firms
would achieve the best collective outcome if they can coordinate with each
other to adopt a mixture of different technologies. To this end, a parameter
$\alpha$ was used to capture the competitiveness level of the product market
and how beneficial it is to achieve coordination, while another parameter
$\mu$ to capture the optimal coordination mixture or diversity of technology
adopters in a group (in the pairwise case, we assume 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 preferences in terms of which outcome should be agreed
upon, thus leading to a larger behavioural space than in the context of
cooperation dilemmas (Han et al.,, 2013, 2017; Han et al., 2015a, ; Sasaki et
al.,, 2015; Hasan and Raja,, 2013). We have shown that whether commitment is a
viable mechanism for promoting the evolution of coordination, strongly depends
on $\alpha$: when $\alpha$ is sufficiently small, prior commitment is highly
abundant leading to significant improvement in terms of social welfare (i.e.
population avarage payoff), compared to when commitment is absent.
Importantly, we have derived the analytical condition for the threshold of
$\alpha$ 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 $\mu$ plays an important role for the
success of commitment strategies as well. In general, when $\mu$ is
intermediate, equivalent to a high level of diversity in group choices,
arranging prior commitments proved to be highly important. It led to
significant 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), we have shown that whenever commitment proposers are
allowed to freely choose which deal to propose to their co-players, our
results show that, in a highly competitive market (i.e. small $\alpha$),
commitment proposers should be strict (i.e. sharing less benefits), while when
the market is less competitive, commitment proposers should be more generous.

In both pairwise and multi-player coordination settings, our analysis has
shown that the cost of arranging agreement must be sufficiently small, to be
justified for the cost and benefit of coordination. This is in line with
previous works in the context of PD and PGG (Han et al.,, 2013, 2017; Han et
al., 2015a, ). It is due to the fact that those who refuse to commit can
escape sanction or compensation. Solutions to this problem have been proposed
in the context of PD and PGG, namely, to combine commitment with peer
punishment, intention recognition, apology or social exclusion to address non-
committers (Han and Lenaerts,, 2016; Han et al., 2015b, ; Han et al., 2015a, ;
Martinez-Vaquero et al.,, 2017; Quillien,, 2020) or to delegate the costly
process of arranging commitment to an external party (Cherry and McEvoy,,
2013; Cherry et al.,, 2017). Our future work will investigate how to combine
prior commitments with such mechanisms to provide a more adaptive and
efficient approach for coordination enhancement in complex systems.

Prior commitments and agreements have been used extensively in the context of
distributed and self-organizing multi-agent systems, for modelling and
engineering a desirable correct behaviour, such as cooperation, coordination
and fairness (Singh,, 1991; Chopra and Singh,, 2009; Winikoff,, 2007). These
works however do not consider the dynamical aspects of the systems nor under
what conditions for instance regarding the relation between costs and benefits
of coordination and those of arranging a reliable commitment, commitment
proposing strategies can actually promote a high level of desirable system
behaviour. Thus, our results provide important insights into the design of
such distributed and self-organizing (adaptive) systems to ensure high levels
of coordination, in both pairwise and multi-party interactions (Bonabeau et
al.,, 1999; Pitt et al.,, 2012).

In future work, we will consider how commitments can solve more complex
collective problems, e.g. in a technological innovation race (Han et al.,,
2020), bargaining games (Zisis et al.,, 2015; Rand et al.,, 2013), climate
change actions (Barrett et al.,, 2007; Santos et al.,, 2020) and cross-sector
coordination (Santos et al.,, 2016), where there might be a large number of
desirable outcomes or equilibriums, especially when the number of players in
an interaction increases (Duong and Han,, 2015; Gokhale and Traulsen,, 2010).

Overall, our work has demonstrated that commitment is a viable tool for
promoting the evolution of diverse collective behaviours among self-interested
individuals, beyond the context of cooperation dilemmas where there is only
one desirable collective outcome (Skyrms,, 1996; Barrett et al.,, 2007). It
thus provides new insights into the complexity and beauty of behavioral
evolution driven by humans’ capacity for commitment (Frank,, 1988; Nesse,,
2001).

##  6 Acknowledgements

T.A.H. is supported by a Leverhulme Research Fellowship (RF-2020-603/9). T.A.H
and A.E. are also supported by Future of Life Institute (grant RFP2-154).

##  7 Appendix

###  7.1 Results for different values of $\theta_{1}$ and $\theta_{2}$

In the main text, we assume that a fair agreement is always arranged. We
consider here what would happen if HP and LP can personalise the commitment
deal they want to propose, i.e. any $\theta_{1}$ and $\theta_{2}$ can be
proposed (instead of always being fair). Namely, Figure 8 shows the average
population payoff varying these parameters, for different values of $\alpha$.
We observe that when $\alpha$ is small, the highest average payoff is achieved
when $\theta_{1}$ is sufficiently small and $\theta_{2}$ is sufficiently
large, while for large $\alpha$, it is reverse for the two parameters. That
is, in a highly competitive market (i.e. small $\alpha$), commitment proposers
should be strict (HP keeps sufficient benefit while LP requests sufficient
payment, from their commitment partners), while when the market is less
competitive (i.e. large $\alpha$), commitment proposers should be more
generous (HP proposes 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 $\epsilon$, $\delta$ and
$\beta$.

Figure 8: Average population payoff as a function of $\theta_{1}$ and
$\theta_{2}$, for different values of $\alpha$ and $\beta$ (for pairwise TD
games). When $\alpha$ is small (panels a and b), the highest average payoff is
achieved when $\theta_{1}$ is sufficiently small and $\theta_{2}$ is
sufficiently large, while for large $\alpha$ (panel c), it is the case when
$\theta_{1}$ is sufficiently large and $\theta_{2}$ is sufficiently small.
Figure 4 also shows that for a small value of $\beta$, the highest average
payoff is achieved when $\alpha$ is very minimal compared to other panels with
higher value of $\beta$ (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: $\delta=4,\ \epsilon=1$; $\beta=0.01,\ 0.1$ and $1$;
population size $Z=100$.

###  7.2 Numerical confirmation of risk-dominant conditions in the N-player
game

See Figure 9 for numerical results confirming the risk-dominant conditions in
the N-player game in the main text.

###  7.3 Results for other intensities of selection in the N-player game

Figure 10 confirms similar observations for other values of intensity of
selection ($\beta$) in the N-player TD game, as compared to Figure 5 in the
main text.

Figure 9: Validation for the analytical conditions under which HP is risk
dominant against strategy HC, LC, HN and LN, see main text. In all cases, with
a small value of $\epsilon$, the HP strategy dominated other players. This
result of this figure is in close accordance with our equations derived above.
Namely, the risk-dominance thresholds of $\epsilon$ for HP (LP) playing
against HC, LC, HN and LN, are, 1.05, 1.31, 12.0 and 58.75, respectively. We
notice a small difference between numerical and theoretical results, since the
latter ones are approximated for larger population sizes. Parameters: in all
panels, $N=5$, $c_{H}=1$, $c_{L}=1$, $b_{H}=6$ (i.e. $b=5$), $\mu=2$, and
$\alpha=0.5$. Figure 10: Frequency of six strategies HP, LP, HN, LN, HC and
LC, as a function of $\alpha$ and for different values of $\epsilon$ and
$\beta$. The commitment proposing strategies HP and LP dominate the population
when the values of $\alpha$ and $\epsilon$ are sufficiently small, in all
cases of $\beta$. Furthermore, as the value of $\epsilon$ 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: $N=5$, $\epsilon=0.1,\ 1,\ 2$; $\beta=0.01,\ 0.1,\ 1$.

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