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Natural games
Jani Anttila a and Arto Annila a,b,c,* ,
a Department of Biosciences, FI-00014 University of Helsinki, Finland
b Institute of Biotechnology, FI-00014 University of Helsinki, Finland
c Department of Physics, FI-00014 University of Helsinki, Finland
ABSTRACT
Behavior in the context of game theory is described as a natural
process that follows the 2nd law of thermodynamics. The rate of
entropy increase as the payoff function is derived from
statistical physics of open systems. The thermodynamic
formalism relates everything in terms of energy and describes
various ways to consume free energy. This allows us to
associate game theoretical models of behavior to physical
reality. Ultimately behavior is viewed as a physical process
where flows of energy naturally select ways to consume free
energy as soon as possible. This natural process is, according to
the profound thermodynamic principle, equivalent to entropy
increase in the least time. However, the physical portrayal of
behavior does not imply determinism. On the contrary,
evolutionary equation for open systems reveals that when there
are three or more degrees of freedom for behavior, the course of
a game is inherently unpredictable in detail because each move
affects motives of moves in the future. Eventually, when no
moves are found to consume more free energy, the extensive-
form game has arrived at a solution concept that satisfies the
minimax theorem. The equilibrium is Lyapunov-stable against
variation in behavior within strategies but will be perturbed by a
new strategy that will draw even more surrounding resources to
the game. Entropy as the payoff function also clarifies motives
of collaboration and subjective nature of decision making.
1. Introduction *
Game theory for the mathematical modeling of human behavior originates from John von Neumann
and John Nash who both drew inspiration from the behavior of thermodynamic systems [1]. Curiously
though applications of game theory in physics and chemistry remain few whereas the breadth of
applications has grown impressive otherwise and covers diverse disciplines, most notably economics
[2], biology [3] and social sciences [4,5] as well as engineering [6], computer and information sciences
[7] and also extends to models of biochemical and biophysical processes [8]. Therefore, when
* Corresponding author.
Email address: arto.annila@helsinki.fi
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considering the broad scope of game theory, could it be, just as the pioneers envisioned, that there is
after all a profound connection between behavior and physical processes via a universal natural law
that underlies the model of behavior in the many forms of games?
The common basic structure of all games implies on one hand that there is some universal principle
underlying behavior of systems. On the other hand, game theory contains a considerable dispersion of
variants, each tailored to model outcomes of specific strategic situations. In particular there is no
consensus about a universal payoff function, denoted pi(s) for a strategy profile s = (s1,...,si), whose
maximization, i.e., payoff pi(s*) for the optimal strategy s*, would cover all incentives of player i in
competition with other players -i. The difficulty in formulating a universal theory of games that would
enclose diverse disciplines seems to relate to what exactly the behavior aims to maximize. In fact,
when considering the variety of circumstances that are confronted during various decision making
processes, it may appear inconceivable that there could possibly be a universal payoff function.
In economics the payoffs are customarily summed up simply as money or, when all other desirables
are included, the payoff is referred to as 'utility' ui(s) or expected utility Ei(s). On the other hand, utility
is an elusive concept, which makes it difficult to assign realistic numerical payoff values. Thus, it
seems problematic to find an accurate and quantitative formulation for all motive forces that underlie
behavior. Moreover, in the light of any chosen payoff, an individual (referred to as a player) appears at
times to act irrationally, e.g., seems to settle for suboptimal personal payoff depending on other
players’ actions, even when considered from the perspective of optimizing over multiple sequential
games. Also, players may seem to act inconsistently with respect to their past actions, sometimes
almost ‘for the sake of it’. Various ways to explain these perplexing findings with bounded rationality
[9] have been pursued. While a consensus is still lacking, the discussion may benefit from the ideas
offered herein.
In biology, too, the choice of payoff is crucial to modeling behavior realistically. The payoff may be
equated with fitness but similarly to utility there is no generally accepted way of quantifying fitness
[10]. Depending on the situation, either reproduction rates or steady-state population densities could be
used, but these may lead to different outcomes [8]. Nevertheless, in the context of behavioral ecology,
an important breakthrough was made when natural selection alone as the profound principle, was
found to prevent alternative strategies from invading a population that is practicing an evolutionarily
stable strategy [11].
In this paper we will re-inspect the theory of games from the physical perspective using the second
law of thermodynamics. The intention here is not to present a new model or to improve on game
theoretical models. Instead, the aim is to map game theoretical concepts into their physical
counterparts and to elucidate that game theory is successful in modeling the behavior of various kinds
of systems because the behavior itself is a physical process that is governed by a universal principle.
This task, despite paralleling the original ideas of von Neumann and Nash, is motivated by the recent
derivation of the principle of increasing entropy as an equation of motion from the statistical physics of
open systems [12]. The revised statistical theory is not limited to closed systems but describes various
natural processes, both inanimate and animate, including behavior of biological, economic and cultural
systems [13,14,15,16,17,18,19,20,21,22]. According to the 2nd law all these systems at various levels
of nature’s hierarchy consume free energy as quickly as possible. This observation leads to the
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proposition that the behavior of many systems, including decision-making processes, could be
described as a natural process so that entropy is the universal payoff function.
2. The Progress of a Game as a Natural Process
The theory of games, as it was formulated by von Neumann, is based on thermodynamics which, in
turn, follows from statistical mechanics. When depicting behavior as a physical process directed by
various incentives, it was not strange for von Neumann, as a physicist, to compare all exchangeable
entities, i.e., assets, in terms of energy and to relate any asset to the average energy density per entity
kBT in the thermodynamic system. It was only when applying the thermodynamic formula to complex
practical systems that von Neumann, in collaboration with economist Oskar Morgenstern abandoned
the principle idea to map one-to-one a physical system to an economical system [1]. When announcing
that money will serve as the payoff, von Neumann and Morgenstern were deeply aware of the
limitations caused by the adopted approximation but could not do better.
Likewise, when Nash formulated the equilibrium concept that carries his name, he adopted the
notion of chemical equilibrium from Gibbs [23]. However, Nash did not aim to make a one-to-one
mapping of mathematical variables to energy densities of chemical compounds but recognized the
resemblance. Nonetheless it is common, especially in economics [24], to see an analogy between the
progress of a chemical reaction toward the thermodynamic equilibrium and the progress of a game
toward a Nash equilibrium. The similarities between mathematical models and physical realizations
may be even more apparent in evolutionary game theory where mixed strategies si are commonly
interpreted as portions of a population expressing a specific behavior. According to this interpretation,
suggested by Nash [25], population densities keep changing, just like reactant concentrations, until a
stable point is reached. The steady state of an ecosystem depends on surrounding conditions, just as the
chemical equilibrium is a function of temperature according to Le Chatelier’s principle [26]. The
stability of a solution condition may also be lost when a new strategy emerges, corresponding to an
addition of a new agent, such as a catalyst, into a chemical reaction mixture. Likewise, the stationary
point may shift when a new species is introduced in an ecosystem. The new species with its
characteristic behavior may use resources that were unreachable to its predecessors. At these critical
events [27], known also bifurcations [28], the old, inferior strategies give way to new, superior
strategies as the system evolves further.
Since von Neumann and Nash, studies in human behavior in various controlled circumstances,
referred to as games, have given rise to stricter and broader solution concepts to classify equilibria as
well as to different kinds of games to model various situations. For example, in the basic zero-sum
game the players exchange assets among each other whereas in a non-zero-sum game they also
compete for external resources. In terms of physics the former variant of the general theme
corresponds to a closed and fixed energy ensemble that behaves as a Hamiltonian system [29] whereas
the latter corresponds to an open system that acquires energy from its surroundings. A game is called
non-generic if a small change to one of the payoffs may remove or add a Nash equilibrium. This means
in terms of physics that such a system is not sufficiently statistical. For example a corresponding
extensive-form game will progress further when a player with a new strategy gains access to additional
assets from the exterior and therefore extending the boundaries of the game. This will increase the
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payoff. In the same way a non-Hamiltonian thermodynamic system will evolve further when a new
reaction component or mechanism gains access to additional free energy. This path-dependent process
will increase entropy which, according to the basic maxim of chemical thermodynamics is equivalent
to the decrease in free energy.
In the quest for a universal theory of games it is of interest that evolutionary game theory
demonstrates scale-independents forms of games [30]. It is successful in explaining various ecological
scenarios where populations compete with each other in the same way as individuals. Moreover,
evolutionarily stable strategies of population games and the Nash equilibria of decision-making games
usually coincide. These examples of scale-free games imply that sentient, population and inanimate
processes are basically alike and operate under a common imperative, only at different scales. Yet
there has been much debate about what exactly the players, may they be molecules, cells, individuals
or populations, aim at maximizing. It seems that a common universal payoff function is required to
unite the diverse models and to place the theory of games on a profound principle. Here this possibility
is examined using statistical physics of open systems.
3. Thermodynamic Formulation of Behavior
The theory of games as it was formulated by von Neumann and expanded by Nash is founded on
Boltzmann’s astounding idea that nature is in motion toward increasingly more probable states.
Boltzmann adopted the probability concept from Descartes, Fermat, Pascal and others who had
computed combinatorial possibilities in the context of gambling but Boltzmann could have also
resorted to the posthumous paper [31] by the Reverend Bayes who had considered circumstantial
possibilities in the context of collecting information [32]. It turns out that new insight to behavior and
the payoff function can be obtained from a re-examination of the probability concept that Boltzmann
placed as the cornerstone of his statistical mechanics.
Boltzmann enumerated, just like counting pips on dice, the isoenergetic configurations that are
commonly referred to as microstates. This invariant probability notion, here referred to as Cartesian, is
constant in energy and thereby it corresponds to stationary systems. Hence the statistical theory, by
founding solely upon this, is limited to changes in configurations of conserved systems. In contrast, the
Bayesian probability P can be seen to vary due to changes in energetic conditions, and thereby it
relates to evolutionary systems. Hence the statistical theory, based on the conditional probability
notion, describes state changes of non-conserved systems. The equation of motion for an evolving
system expresses the principle of increasing entropy
ln
tBtt
j k
4
,
0 .
j jkB
d S k dP d N AT k L
= = -=³
å
(1)
In the context of game theory the rate of entropy increase dtS is the payoff function that values the
outcomes of various jk-transactions where k-assets of players are transformed to j-assets of other
players (and vice versa). For example, a player uses money in his possession to buy goods from
another. The actions bring about changes dtNj in the j-assets of the player concurrently with changes
dtNk in the k-assets of the other player. In fact not only money and goods are rated by energy
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differentials but literally everything is valued in terms of energy differentials. Therefore the holistic
formalism is able to describe the behavior of complex systems just as of simple systems. The quest to
consume free energy in least time is ubiquitous and independent of mechanisms. The overall sequence
of transactions, referred to as the course of a game, advances move-by-move when the free energy Ajk
= Dmjk – iDQjk is consumed. The term Dmjk = mj – Smk contains the scalar potential differences in the
assets where mj = kBTln[Njexp(Gj/kBT)] is a function of the number Nj and value Gj of the j-asset. When
the driving force contains only the potential differences, the system is closed to net flows of energy
from its surroundings. In the corresponding zero-sum game the transactions between players bring
about merely an exchange of assets but the total status over all players remains invariant. The invariant
nature of zero-sum games is stated by the minimax theorem [33]. The term iDQjk in Eq. 1 is the energy
influx from the surroundings to an open, extensive-form game. This term imposed by the surroundings
distinguishes the statistical physics of open systems from the conventional formalism used by
Boltzmann and others ever since. The influx, which is often equated with income, is incorporated in
the assets by the players’ actions. Conversely, a player may lose his assets to the others as well as to
the surroundings by misfortunate moves.
A move by a player will cause a change in assets. The rate of change is proportional to the driving
force Ajk by the coefficient sjk [13]
d N
s
5
.
tj jkjkB
k
A k T
= -å
(2)
When Ajk < 0, the jk-transaction will increase dtNj > 0, and vice versa. The particular functional form in
Eq. 2 ensures that conservation of energy is satisfied in every move [12]. When Eq. 2 is inserted in Eq.
1, entropy is found to increase almost everywhere dtS ≥ 0 because each square Ajk2 ≥ 0. This is the
principle of increasing entropy.
The proportionality coefficient sjk represents a particular mechanism that channels the jk-
transaction. For example, a more effective means of trading brings about faster changes in the assets.
The rate is not immaterial because the driving force (Ajk) is, in turn, a function of assets (mj). In other
words, behavior and the motives of behavior are inseparable from each other. Indeed, it is witnessed
that when stakes are raised, behavior will change. For example, in the well known ultimatum game the
probability of rejecting a share, whether fair or unfair, decreases with the increase of the absolute
amount offered [34]. However, in classical game theory the amount of assets in players’ possession is
customarily ignored.
According to the physical portrayal of games more effective mechanisms are favored by the
transactions themselves as they allow for a faster maximization of entropy. The quest to increase
entropy in the least time is also known by the maximum entropy production principle [35]. According
to the adopted self-similar thermodynamic formalism a game itself is also a mechanism that may
evolve further to facilitate the overall consumption of free energy. In other words there are games
being played within games, in accordance with hierarchical system theory [36]. For example, the
behavior of a citizen amongst others can be regarded as a game that ultimately also contributes to
international relations that, in turn, can be regarded as a game among nations. In the quest for the
maximal dispersal of energy the systems will form a coalition, i.e., a larger system where it is more
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effective in acquiring and consuming free energy and distributing the acquired flows among its
constituent systems [37].
Moreover, the functional form of dtS in Eq. 1 implies that the particular possessions of a player
affect not only their decisions but those of others as well. The product form of Eq. 1 states that actions
(dtN) are inseparable from possessions (Ajk). This interdependency is the source of unpredictability
when an open, extensive-form game is played by three or more parties [38]. The courses of non-
deterministic games vary and they do not necessarily end up with the same outcome because a move at
any stage depends on past moves and conversely restricts the future choice of moves (Fig. 1). At the
branching points, to be precise, the derivate đtS (Eq. 1) is inexact. In terms of physics the game is a
non-Hamiltonian system with three or more degrees of freedom where the driving forces and energy
flows are inseparable [12,39]. This thermodynamic interdependency between flows of energy and the
free energy that drives all other flows underlies the interdependency between the strategy si and its
complement, s-i, the strategies played by all other players. In other words, the thermodynamic theory
gives the reason why the decision made by a player is dependent on the decisions of other players,
which, in turn, are dependent on the first player’s decision. In fact repeatedly changing conditions may
drive repeated changes of strategies. For example, it has been shown that in a well studied game, the
iterated continuous prisoner’s dilemma, no strategy is evolutionarily stable [40]. Moreover, the
intractability of extensive-form games is understood as an inherent characteristic of open systems.
Owing to the net influx or efflux of energy to or from the system, there is no norm and hence no
unitary transformation either to obtain a solution or to predict the trajectory toward the solution
concept. Even a minor move will perturb the energy content of the system and hence affect the future
course. This is characteristic of chaotic games [41]. The sequence of moves, i.e., kinetics is, instead of
using Eq. 2, often modeled by the deterministic law of mass action [42], but then thermodynamics and
kinetics become incompatible with each other [12]. Alternatively, a sequence of moves is modeled as a
Markov chain [43]. However, even when present probabilities depend on the past, the probabilities are
not understood as physical.
At any given time the game, described as a natural process, can be assigned with the additive
logarithmic status measure known as entropy [12]
å
Õ
6
ln ln1
BBjBj jkB
jk
j
SkPkPkNA k T
æ
ç
ç
è
ö
÷
÷
ø
==»-
å
(3)
over the product of probabilities Pj. Entropy will increase toward the steady state value Smax where all
free energy terms Ajk = 0. In this case no strategy, i.e., no choice of jk-moves can be found by any
player to improve the distribution of assets Nj among the players or to acquire more assets from the
surroundings. Customarily, the optimal behavior is referred to as the evolutionary stable strategy
(ESS). The free energy minimum state can be proven stable against perturbations dNj using Eqs. 1 and
3 in the Lyapunov criteria S(dNj) < 0 and dtS(dNj)/dt > 0 [44,45]. In other words, there is no action that
could improve the status and no strategy that any player could play that would return a higher payoff,
given all the strategies played by the other players.
The steady-state partition of assets is given by the condition dtS = 0 which yields from Eq. 1
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7
()
jk jkB
G i Q
- D
k T
jk
k
N N e
- D
=Õ
(4)
This is the condition of reaction equilibrium [26]. The dispersal of assets at the free energy minimum
is a skewed, nearly log-normal distribution [14]. Indeed, studies of behavior in many-players games
reveal that the strategies of individuals will evolve and the game will end up in a stationary state where
only few have gathered large assets while most players must have settled for moderate possessions and
relatively few have only little. Both the maximum and the amount of skew for a specific situation
depend on the overall energy content of the system (Fig. 1). This is familiar from the distributions of
ecosystems [46] and economic systems [47] and from partitions of elementary chemical and physical
systems [48].
Figure 1. Extensive-form game branching on the left side is depicted on the right
side as a natural process that brings about changes in diverse assets, indicated by the
solids, that correspond to populations Nj on energy levels Gj. As denoted, move by
move, by arrows in the sequence of state changes (A – D) the payoff function as
entropy increases until a solution concept corresponding to a stationary state of free
energy minimum has been attained. This maximum entropy partition as a
thermodynamic steady state is an evolutionary stable solution concept where the net
dissipation (wavy arrows) vanishes. No move is able to bring more assets, i.e.,
energy to the process that settles to the evolutionary stable state.
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8
The optimal distribution given by Eq. 4 is the result of a natural process just as it is the outcome of
an extensive-form game. However, it is worth emphasizing that at the steady state there are no net
fluxes to or from the systems just as the players’ total assets at a particular solution concept are no
longer changing. This parallels the fixed-energy condition of stationary systems. It means that even
though the outcomes of repeated games differ, each of them is a feasible solution concept, i.e., the
minimax condition [1] just as in a zero-sum game is satisfied for all players. Also, the proof of the folk
theorem [2] rests on the invariant nature of an outcome [49].
4. Motives of Behavior
Is it reasonable, as argued above, to equate incentives of behavior with the imperative to increase
entropy? Would not the association of animate actions with the physical principle of decreasing free
energy quench all degrees of freedom from behavior? These and other concerns about the use of rate of
entropy increase as the universal payoff function deserve to be addressed.
According to the naturalistic tenet, behavior is, just like any other natural process, limited by the
free energy because a more generous stance would violate the conservation of energy. In other words,
the free energy sums up all resources that an individual or any other agent has in possession or access
to with its available mechanisms. Beyond that no one can act. This superior role of free energy is
apparent, for example, from actions that a social system is able to take when fostering its subordinate
individuals whereas an individual, even a rich one, rarely has enough power to act against society for
any significant periods of time. The thermodynamic description of behavior as an energy transduction
process leaves all of the available pathways open for conduction, but there is the natural bias for the
best path, known also as a geodesic, that will diminish free energy as soon as possible [50]. Staying on
an optimal trajectory requires both incessant evaluation of alternative pathways and redirection after
each choice. In other words the game is changing as it is played. Thus behavior cannot be reduced to
precisely predictable sequence of acts.
The rate of entropy increase in the least time (Eq. 1) as the motive of behavior measures all
available sources of free energy (Ajk) weighted by rates of their consumption (dtNj). In this sense
entropy, just like utility, considers numerous variables that influence behavior. However, since the
payoff function of the rate of entropy increase evaluates everything in terms of energy, it makes all
motives of behavior commensurable with each other. In practice, though, it may be difficult to
accurately assign an energetic value to every option, but in a statistical sense the behavior itself reveals
the current value of a strategy. Nonetheless, it is legitimate to question whether a single number can
possibly sum up all, at times even conflicting, motives of behavior and eventually to anticipate
behavior. The answer is yes only in the statistical sense. The statistical notion is applicable when any
one move will not change the course of game too much. In physical terms, the condition is valid when
the consumption of Ajk is small in comparison to the average energy content in the system kBT.
However, events where Ajk ≈ kBT, may be infrequent, but they do happen. This long-tail trait of a
probability distribution is characteristic of natural processes [51]. When S = kBlnP (Eq. 3) is not a
sufficient statistic for kBT, the state of a game is given best by the probability [12,39]
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()
!.
j
jk jkB
N
G i Q
- D
k T
jkj
jjk
PP N eN
- D
æ
ç
ç
è
ö
÷
÷
ø
==
Õ Õ Õ
(5)
The exponential form reveals that any one Pj, for which Ajk = Dmjk – iDQjk t kBT, contributes
substantially to P. The jk-move corresponding to the consumption of Ajk will change Pj significantly,
i.e., beyond the statistical approximation lnNj! ≈ NjlnNj – Nj which is applicable to large, quasi-
stationary populations. In this case the probability of a small system will not evolve smoothly but
moves in steps according to dtP = LP (Eq. 1). In practice this means that the rate of entropy increase
cannot be used to extrapolate a specific scenario. For example, when an individual labeled with j
happens to strike a particularly prosperous deal, his status measured by Pj will step up abruptly.
Nevertheless, the status P of the overall course of the game, comprising many players, will remain
sufficiently statistical and it will not be affected all that much by a single move but moves smoothly.
The description of a game as a natural process is holistic so that any jk-move will, in principle,
affect the status of any other player. The physical probability in Eq. 5 defines, by jk-indexing, the
interdependency among the densities-in-energy. A pair or a set of decisions are referred to as strategic
complements when they are constructively reinforcing one another and as strategic substitutes when
they are destructive in offsetting one another. The course of a game depends on coherent moves, which
in terms of physics, means that the flows of energy interfere with each other when the affine, curved
energy landscape is in evolution toward the stationary-state flatness [52,53]. Both sequential and
simultaneous moves are accommodated in the formulation (Eq. 1) but since velocities of energy flows
are limited, ultimately by the speed of light, only the sequential actions display causality [39] to affect
subsequent decisions whereas simultaneous moves are independent, which is a familiar notion in
sealed first-price auctions. Moreover, individual behavior via social interactions has been understood
as the mechanism that bonds together the affine energy landscape and generates its evolution [54].
The physical probability that is conditioned in energetic terms (Eq. 5) clarifies also why mimicking
(imitation) is often a successful strategy. A priori, i.e., at an initial state it may not be obvious which
particular move will consume free energy in the least time but pioneers will search for paths, e.g., by
trial and error. Initially the optimality of a path is less important because just about any strategy will
produce entropy. Moreover since no experience, i.e. references, has been accumulated, the optimality
cannot be assessed. Therefore a successor, when mimicking the established behavior, will follow, if
not the best, at least a reasonable trail formed by the path breakers. Explorations are per definition
suboptimal moves. This is consistent with rational ignorance [55] which states that the act of acquiring
information on the best possible strategy or path may be too costly compared to expected and uncertain
benefits to the player. The thermodynamic theory shows that the mere move to set a path will change
the setting for subsequent moves as well [56]. The probabilities of future decisions are affected by past
acts, in other words a specific state of a game depends on its history. The conditional interdependence
among strategies is also familiar from cemented suboptimal standards. It is tedious to improve a
widely adopted standard simply because initial payoff will suffer from the limited scope of
applicability of the reform. Conversely, marginal benefits for conformists are easily available, whereas
significant gains are in the sight of a rebel. The conventional way of thinking as an established means
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of energy dispersal is preferred rather than making much additional effort to explore beyond
paradigms. Players are motivated to explore new strategies when they see a possibility for greater
payoffs, i.e., a larger perceived gradient in free energy than is consumed by the study itself. New
strategies may emerge from intentional manipulation or sporadic fluctuations also known as random
variations. For the payoff, i.e. the entropy increase, it does not matter whether the move is intentional
or accidental, since only the outcome is valued. The changes may be, for example, mutations in a
biological context or reorganizations in the brain in a decision making context. When a game is
maturing toward a solution concept, most new strategies are not so successful, but some may still tap
into potentials better than competitors and gain ground.
5. Discussion
Game theory accounts for behavior in remarkably diverse circumstances, yet it is pertinent to ask
what behavior actually is. Game theory rationalizes behavior by modeling it as a game that aims at
maximizing payoff. However, the nature of a game as a process and its objective has remained
obscure. Here games are described as natural processes that increase entropy in the least time. This
tenet, while founded on the 2nd law of thermodynamics as the universal principle, may at first appear
superficial and deficient as if it was neglecting important factors such as the role and asymmetric
distribution of information among players. However, the naturalistic view is holistic in relating
everything to everything in terms of energy. As the flow of energy is the sole means of conveying
information, this means that a piece of information is also an asset. This stance is valid because any
form of information is bound to its physical representation [57,58,59], which, in turn, is subject to the
laws of thermodynamics. Furthermore, the accumulation of information, e.g., a learning process itself,
can be understood as a natural process of formation and changing of paths (geodesics) for flows of
energy that represent information. Therefore, the value of information gained by behaving in a certain
way, i.e., by playing a certain strategy which acquires information for the future accumulation of assets
has to be accounted for. Therefore a change in strategy is a natural consequence of accumulating assets
because the acquired assets open new opportunities for the reduction of free energy. The acquisition of
information about other players’ types, e.g., to account for a Bayesian game [60], is contained in the
physical formulation of games as natural processes. This physical correspondence is also reflected in
the purification theorem. It states that mixed strategy equilibria can be obtained as the limit of pure
strategy equilibria from a perturbed game of incomplete information. Physically speaking, a mixed
state can be constructed as the limit of pure states that are perturbed by energy in mutual interactions.
Information is energy in interactions, when defined in thermodynamic [58] rather than mathematical
[61] terms.
The intimate interdependency between behavior and its motives, that are, physically speaking,
flows of energy and free energy, is apparent when acquired knowledge is used to anticipate moves by
others. The extensive-form game where knowledge is accumulated is directed toward a self-confirming
equilibrium [62] just as a natural process spontaneously progresses toward a stable state. An extensive
series of repeated games will eventually reveal, in mutual transactions, all characteristics of all players.
Thus all conceivable paths of actions are open to maximize the total payoff. This revelation shifts the
steady state from the Bayes-Nash equilibrium to the ultimate optimum [63,64]. This tedious
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optimization procedure relates to computational complexity [65]. There is no algorithm that would
solve this non-deterministic polynomial time problem [66]. At the Lyapunov-stable state there are no
unexposed assets and no strategies that could possibly shift the equilibrium any further.
The extensive-form game models the fact that behavior depends on surrounding conditions. A
change in the surroundings relates to changes in the values of a payoff matrix, and more efficient
strategies to which higher payoffs are available may emerge. Any such change will bring instability to
the game. The approach of the new equilibrium may even be chaotic because any early move will
affect the set of moves available in the future. Moreover, when a particularly effective strategy is
found and executed, it may consume assets more rapidly than they are replenished. Consequently, after
a period of overexploitation, the system must retract and abandon the once so lucrative, but
unsustainable strategy. For this reason animal populations oscillate and economic cycles follow one
another.
The connection between the entropy maximum and game theoretical equilibrium is, as such, not a
novel proposition [67]. However, here the rate of entropy increase is provided in a mathematical form
that is equivalent to the rate of free energy decrease. This is essential. Since everything can be valued
in terms of energy, the rate of entropy increase qualifies as the universal payoff function. Moreover,
statistical physics of open systems links the principle of increasing entropy to the principle of least
action which guides processes along the optimal paths that bring the system to the stationary state in
the least time. Admittedly, just like free energy, entropy in a complicated system is a function of many
variables that are indexed by j and k. In this way entropy as the additive figure of merit is of course
very much in line with observations that human and other animate behavior is difficult, if not
impossible, to account solely on the basis of a single motive. While justified by the profound principle
it may be tedious to expand the entropy function in every detail to model practical situations. Rather it
would seem sensible to model only the terms that are anticipated to be significant in decisions that are
confronted in specified circumstances. However, thermodynamics clarifies that no precise predictions
are available even from very detailed formulation because the moves themselves will change the
conditions and prompt different moves. Moreover, the physical portrayal of games as natural processes
by the rate of entropy increase as the payoff function gives justification for mixed and varying
strategies over pure and fixed strategies. Diversity in behavior, like biodiversity, allows the entire
system to consume free energy more and faster than would be possible via individual and invariant
strategies.
Entropy as the payoff function also clarifies the subjective nature of decision making when
choosing a strategy because payoff is a function of the possessions and moves that are available for a
specific player. Moreover, in hopes of making a rational choice the player, who is equipped with
appropriate knowledge, may discount a future payoff when making present-day decisions. Thus there
is no universal rational choice, which is why some actions may seem irrational to an outside observer.
In this sense the thermodynamic theory addresses some of the critical concerns about rational choice
[68]. Furthermore, physical formalism does not allow for observations without interactions. Therefore
an observer will inevitably affect, i.e., integrate himself in, the course of a game. Common values are
approached via integration where superior free energy possessions and effective consumption
strategies acquired by energy-intense players tend to impose on others what is deemed as rational,
when in fact much of this “rationality” is in fact mimicry and submission to authority.
11
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The physical portrayal of games as natural processes illuminates not only competition over assets
but also cooperation among individuals. A coalition is regarded as a strategy, just as any other
mechanism to increase entropy. The group possesses more means and more assets to access higher
status in entropy than any one individual could master independently. The consensus in decision
making is motivated only if it provides the means for each individual to attain a higher entropic status
than would be available by independent moves. It is not unusual that, when circumstances change,
coalitions will expire or reorganize to adapt to the new circumstances. This understanding adds to the
ongoing debate concerning the emergence of cooperation, to which several solutions have been
proposed [69,70]. Parameterization of models with a physical quantity may help to distinguish the type
of game, for example snowdrift or prisoner’s dilemma, to represent a situation.
Finally, the tragedy of the commons [71] that has also been analyzed by game theory [72] deserves
clarification. The detrimental scenario that is driven by short-sighted individual incentives continues
when resources and means of social bonding are insufficient. This alerting sequence of moves toward
ruination is understood by the thermodynamic theory as probable. According to the natural law, when
energy in the surrounding supplies falls, due to exploitation by individuals, below that contained in the
social system, the flow of energy is redirected according to the 2nd law away from the society to the
surroundings. Consequently, the social system keeps draining its cohesion just when it would
desperately need more energy to re-establish vital mechanisms such as social bonding to enforce co-
operation that would be necessary for society to behave in a sustainable manner.
Acknowledgements
We thank Niall Douglas, Vesa Kanniainen and Leena Kekäläinen for inspiring comments and
valuable corrections.
References
1. von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior. Princeton University Press: NJ, 1944.
2. Fudenberg, D.; Tirole, J. Game Theory. The MIT Press: Massachusetts, 1991.
3. Maynard Smith, J. Evolution and the Theory of Games. Cambridge University Press: Cambridge, UK, 1982.
4. Colman, A.M. Game Theory and its Applications in the Social and Biological Sciences. Butterworth-Heinemann:
Oxford, UK, 1995.
5. Colman, A.M. Cooperation, psychological game theory, and limitations of rationality in social interaction.
Behavioral and Brain Sciences 2003, 26, 139–198.
6. Srivastava, V.; Neel, J.; Mackenzie, A.B.; Menon, R.; Dasilva, L.A.; Hicks, J.E.; Reed, J.H.; Gilles, R.P. Using
game theory to analyze wireless ad hoc networks. Communications Surveys and Tutorials, IEEE 2005, 7, 46–56.
7. Nisan, N.; Roughgarden, T.; Tardos, E.; Vazirani, V. (Eds.) Algorithmic Game Theory. Cambridge University
Press: Cambridge, UK 2007.
8. Schuster, S.; Kreft, J.; Scrhoeter, A.; Pfeiffer, T. Use of game-theoretical methods in biochemistry and biophysics, J.
Biol. Phys. 2008, 34, 1–17.
9. Gigerenzer, G.; Selten, R. Bounded Rationality: The Adaptive Toolbox. MIT Press: Cambridge, Mass, 2002.
10. Grafen, A. The formal Darwinism project: a mid-term report. J. Evol. Biol. 2007, 20, 1243–1254
11. Maynard Smith J.; Price G.R. The logic of animal conflict. Nature 1973, 246, 15–18.
12. Sharma, V.; Annila, A. Natural process – Natural selection. Biophys. Chem. 2007, 127, 123–128.
(doi:10.1016/j.bpc.2007.01.005)
13. Jaakkola, S.; Sharma, V.; Annila, A. Cause of chirality consensus. Curr. Chem. Biol. 2008, 2, 53–58.
Page 13
(doi:10.2174/187231308784220536) (arXiv:0906.0254)
14. Grönholm, T.; Annila, A. Natural distribution. Math. Biosci. 2007, 210, 659–667. (doi:10.1016/j.mbs.2007.07.004)
15. Würtz, P.; Annila, A. Roots of diversity relations. J. Biophys. 2008, (doi:10.1155/2008/654672) (arXiv:0906.0251)
16. Annila, A.; Annila, E. Why did life emerge? Int. J. Astrobiol. 2008, 7, 293–300. (doi:10.1017/S1473550408004308)
17. Karnani, M.; Annila, A. Gaia again. Biosystems 2009, 95, 82–87. (doi: 10.1016/j.biosystems.2008.07.003)
18. Sharma, V.; Kaila, V.R.I.; Annila, A. Protein folding as an evolutionary process. Physica A 2009, 388, 851–862.
(doi:10.1016/j.physa.2008.12.004)
19. Würtz, P.; Annila, A. Ecological succession as an energy dispersal process. Biosystems 2010, 100, 70–78.
20. Annila, A.; Salthe, S. Physical foundations of evolutionary theory. J. Non-equilb. thermodyn. 2010, 35, 301–321.
21. Annila, A.; Salthe, S. Economies evolve by energy dispersal. Entropy 2009, 11, 606–633.
22. Annila, A.; Salthe, S. Cultural naturalism. Entropy 2010, 12, 1325–1343.
23. Gibbs, J.W. The Scientific Papers of J. Willard Gibbs. Ox Bow Press: Connecticut, 1993–1994.
24. Samuelson, P. Foundations of Economic Analysis. Harvard University Press: Cambridge, MA, 1947.
25. Nash, J.F. Non-cooperative games. Ph.D. Dissertation. Princeton University: NJ, 1950.
26. Atkins, P.W.; de Paula, J. Physical Chemistry. 8th ed. Oxford University Press: New York, NY, 2006.
27. Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality: an explanation of 1/f noise. Phys. Rev. Letters 1987, 59,
381–384.
28. May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459–467.
29. Hofbauer, J. Evolutionary dynamics for bimatrix games: A Hamiltonian system? J. Math. Biol. 1996, 34, 675–688.
30. Weibull, J.W. Evolutionary Game Theory. MIT Press: Cambridge, MA, 1995.
31. Bayes, T. A letter to John Canton. Phil. Trans. R. Soc. 1763, 53, 269–271.
32. Jaynes, E.T. Probability Theory: The Logic of Science. Cambridge University Press: Cambridge, UK, 2003.
33. von Neumann, J. Zur Theorie der Gesellshaftsphiele. Math. Ann. 1928, 100, 295–320.
34. Cameron, L.A. Raising the stakes in the ultimatum game: experimental evidence from
Indonesia. Economic Inquiry 1999, 37, 47–59.
35. Martyushev, L.M.; Seleznev, V.D. Maximum entropy production principle in physics, chemistry and biology.
Physics Reports 2006, 426, 1–45.
36. Salthe, S.N. Summary of the principles of hierarchy theory. General Systems Bulletin 2002, 31, 13–17.
37. Annila, A.; Kuismanen, E. Natural hierarchy emerges from energy dispersal. BioSystems 2009, 95, 227–233.
doi:10.1016/j.biosystems.2008.10.008
38. Broom, M.; Cannings, C.; Vickers, G.T. Multi-player matrix games. Bull. Math. Biol. 1997, 59, 931–952.
39. Tuisku, P.; Pernu, T.K.; Annila, A. In the light of time. Proc. R. Soc. A. 2009, 465, 1173–1198.
doi:10.1098/rspa.2008.0494
40. Lorberbaum, J. No strategy is evolutionarily stable in the repeated prisoner’s dilemma. J. Theor. Biol. 1994, 168,
117–130.
41. Salvetti, F.; Patelli, P.; Nicolo, S. Chaotic time series prediction for the game, Rock-Paper-Scissors. Appl. Soft
Comput. 2007, 7, 1188–1196. http://dx.doi.org/10.1016/j.asoc.2006.01.006
42. Waage, P.; Guldberg, C.M. Studies concerning affinity. Forhandlinger I, 35 Videnskabs-Selskabet: Christiania,
1864.
43. Parthasarathy, T., Stern, M. Markov Games: a Survey, Differential Games and Control Theory II p. 1–46, Edited by
E.O. Roxin et al, Marcel Dekker: New York, NY, 1977.
44. Kondepudi, D.; Prigogine, I. Modern Thermodynamics. Wiley: New York, NY, 1998.
45. Strogatz, S.H. Nonlinear Dynamics and Chaos with Application to Physics, Biology, Chemistry and Engineering.
Westwiev: Cambridge, MA, 2000.
46. Rosenzweig, M.L. Species Diversity in Space and Time. Cambridge University Press: Cambridge, 1995.
47. Pareto V. Manuale di economia politica. CEDAM: Padova, 1974; Manual of political economy. Translation by
Page, A.N.; A.M. Kelley; New York, NY, 1971.
48. Newman, M.E.J. Power laws, Pareto distributions and Zipf's law. Contemporary Physics 2005, 46, 323–351.
49. Rubinstein, A. Equilibrium in Supergames with the Overtaking Criterion. J. Econ. Theory 1979, 21, 1–9.
doi:10.1016/0022-0531(79)90002-4
50. Kaila, V.R.I.; Annila, A. Natural selection for least action. Proc. R. Soc. A. 2008, 464, 3055–3070.
doi:10.1098/rspa.2008.0178
51. Anderson, C. The long tail. Wired 2004, Oct.
13
Page 14
52. Annila, A. Space, time and machines. 2009, arXiv:0910.2629.
53. Annila, A. The 2nd law of thermodynamics delineates dispersal of energy. Int. Rev. Phys. 2010, 4, 29–34.
54. Marvel, S.; Strogatz, S.; Kleinberg, J. Energy landscape of social balance. Phys. Rev. Lett. 2009, 103, 19701.
55. Bromberger, S. Rational ignorance. Synthese 1988, 74, 47–64.
56. Annila A. All in action. Entropy 2010, 12, 2333–2358..
57. Landauer, R. Information is physical. Physics Today 1991, May.
58. Karnani, M.; Pääkkönen, K.; Annila, A. The physical character of information. Proc. R Soc. A. 2009, 465, 2155–
2175.
59. Herrmann-Pillath, C. Entropy, Function and Evolution: Naturalizing Peircian Semiosis. Entropy 2010, 12, 197–242.
60. Harsanyi, J.C. Games with incomplete information played by “Bayesian” players, I-III. Management Science 1967,
14, 159–182.
61. Shannon, C.E.; Weaver, W. The mathematical theory of communication. The University of Illinois Press: Urbana,
IL, 1962; Shannon, C.E. The mathematical theory of communication. Bell System Technical Journal 1948, 27, 379–
423, 623–656.
62. Fudenberg, D.; Levine, D.K. Self-confirming Equilibrium. Econometrica 1993, 61, 523–545.
63. Gibbard, A. Manipulation of voting schemes: a general result. Econometrica 1973, 41, 587–601.
64. Holmström, B. On incentives and control in organizations. Ph.D. thesis, Stanford University: CA, 1977.
65. Sipser, M. Introduction to the theory of computation. Pws Publishing: New York, NY, 2001.
66. Annila, A. Physical portrayal of computational complexity. 2009 (arXiv:0906.1084)
67. Topsøe, F. Entropy and equilibrium via games of complexity. Physica A 2004, 340, 11–31.
68. Hargreaves-Heap, S.P.; Varoufakis, Y. Game Theory: a Critical Introduction. Routledge: New York, NY, 1995.
69. Nowak, M.A. Five rules for the Evolution of Cooperation. Science 2006, 314, 1560–1563.
70. West, S.A.; Gardner, A.; Shuker, D.M.; Reynolds, T.; Burton-Chellow, M.; Sykes, E.M.; Guinnee, M.A.; Griffin,
A.S. Cooperation and the scale of competition in humans. Curr. Biol. 2006, 16, 1103–1106.
71. Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248.
72. Hardin, G. Filters Against Folly, How to Survive despite Economists, Ecologists, and the Merely Eloquent. Viking
Penguin: New York, NY, 1985.
14
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