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- (10) Supplementary material.
- (11) While our work is among first applications of game theory to statistical physics, connections between mathematical economics, thermodynamics and econophysics are well-known saslow ; candeal ; tsirlin ; smith ; yakovenko ; bruce ; e.g. there are interesting analogies between the axiomatic features of entropy and utility candeal . For application of statistical physics idea in evolutionary game theory see chatelier .
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- (23) Axiomatic approaches to equilibrium thermodynamics have a long history; see callen ; buchdahl ; cooper ; lieb . They revolve around axiomatic introduction of entropy, whereas in our situation the entropy is just assumed to hold its standard form.
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I Supplementary Material
i.1 1. Calculation of the minimum entropy for a fixed average energy
Here we show how to mimimize entropy
for a fixed average energy
Energy levels are given.
Since is concave, its minimum is reached for vertices of the allowed probability domain. This domain is defined by the intersection of (17) with probabilities that support constraint (18). Put differently, as many probabilities nullify for the minimum of , as allowed by (18). Hence at best only two probabilities are non-zero.
We now order different energies as
and define entropies , where only states and with have non-zero probabilities:
where is the step-function ( and ), and where we assume . Likewise, for :
|Bargaining theory||Statistical mechanics|
|Utilities of players||Entropy and minus energy|
|Joint actions of players||Probabilities of states for the physical system|
|Feasible set of utility values||Entropy-energy diagram|
|Defection point||Initial state|
|Pareto set||Maximum entropy curve for|
|positive inverse temperatures|
Utilities (payoffs) are normally dimensionless and are defined subjectively, via preferences of a given agent (player) luce-1 ; rubin ; roth_book-1 ; karlin-1 . Entropy and energy are physical, dimensional quantities grandy-1 ; balian-1 . The player’s utility ( denote the first and second player, respectively) depends on the actions and taken by the first and second player respectively luce-1 ; rubin ; roth_book-1 . Entropy and energy depend on the probability of various states of the physical system grandy-1 ; balian-1 . Unlike actions, these probabilities do not naturally fraction into two different components. Hence it is unclear how to apply the non-cooperative game theory to statistical mechanics. For cooperative game theory this problem is absent, since the actions are not separated.
All utility values from the feasible set are potential outcomes of bargaining. The feasible set is normally convex and it is even modified to be comprehensive luce-1 ; rubin ; roth_book-1 , i.e. if belongs to the feasible set, then all points with and also belong to the feasible set. The features of comprehensivity is motivated by the observation that (arbitrary) worse utility values can be added to the existing feasible set rubin ; roth_book-1 . The observation is not at all obvious (or innocent) even in the game-theoretic context luce-1 .
The entropy-energy diagram is a well-defined physical object that does not allow any (more or less arbitrary) modification. It is not convex due to the minimal entropy curve; see Fig. 5.
Game-theoretically, the defection point is a specific value of utilities which the players get if they fail to reach any cooperation and/or agreement luce-1 ; rubin ; roth_book-1 . Normally, the defection point corresponds to guaranteed payoffs of the players. The defection point does not have any direct physical analogy. Instead of it we need to employ the notion of the initial point that as such does not have game-theoretic analogies (at least within axiomatic bargaining).
For a give set of utilities belonging to the feasiblity set , the Pareto set is defined as a subset of such that if there does not exist any with and , where at least one of inequalities is strict luce-1 ; rubin ; roth_book-1 . Thus there are no utility values that are jointly better than any point from the Pareto set.
If vary over a convex set, and if both and are concave functions of , then every point from can be recovered from conditional maximization of with a fixed or vice versa karlin-1 (Karlin’s lemma).
Within statistical physics the Pareto line coincides with the branch of the maximum entropy curve with a positive inverse temperatures , because this branch is also the minimum energy curve for a fixed entropy. Both entropy and energy are concave functions of probability, and the probability itself is defined over a convex set (simplex). Hence Karlin’s lemma applies.
By analogy to the Pareto set we can define also the anti-Pareto set, which (as the opposite to the Pareto set) relates to “worst” points of . The anti-Pareto set is defined as a subset of such that if there does not exist any with and , where at least one of inequalities is strict. On the entropy-energy diagram the only anti-Pareto point coincides with . After imposing Axiom 1 and going to the new set of feasible states, the unique anti-Pareto point coincides with the initial state.
i.2 2. Features of the Nash solution (9)
i.2.1 2.1 Concavity
where (for obvious reasons) the maximization was already restricted to the maximum entropy curve . Now recall that is a concave function. Local maxima of are found from :
We shall now show that this local maximum is unique and hence coincides with the global maximum. For any concave function we have for
which produces after re-working and using (25) for :
Hence is the unique global maximum of .
i.2.2 2.2 Comments on the textbook derivation of (9).
In the main text we emphasized that the derivation of the solution (9) for the axiomatic bargaining problem that was proposed by Nash nash-1 and is reproduced in textbooks luce-1 ; rubin ; roth_book-1 has a serious deficiency. Namely, (9) is derived under an additional assumption, viz. that one can enlarge the domain of allowed state on which the solution is searched for. This is a drawback already in the game-theoretic set-up, because it means that the payoffs of the original game are (arbitrarily) modified. In contrast, restricting the domain of available states can be motivated by forbiding certain probabilistic states (i.e. joint actions of the original game), which can and should be viewed as a possible part of negotiations into which the players engage. For physical applications this assumption is especially unwarranted, since it means that the original (physical) entropy-energy phase diagram is arbitrarily modified.
We now demonstrate on the example of Fig. 4 how specifically this assumption is implemented. Fig. 4 shows an entropy-energy diagram in relative coordinates (8) with being the maximum entropy curve. The affine transformation were chosen such that the Nash solution (9) coincides with the point . Now recall (25). Once , then , and since is a concave function, then all allowed states lay below the line ; see Fig. 4. If now one considers a domain of all states (excluding the initial point ) lying below the line (this is the problematic move!), then is the unique solution in that larger domain. Moving back to the original domain and applying the contraction invariance axiom, we get that is the solution of the original problem.
i.3 3. Maximum entropy method and its open problem
i.3.1 3.1 Maximum entropy method: a reminder
The method originated in the cross-link between information theory and statistical mechanics jaynes-1 . It applies well to quasi-equilibrium statistical mechanics grandy-1 ; balian-1 , and developed to become an inference method (for recovering unknown probabilities) with a wide range of applications; see e.g. grandy-1 ; balian-1 ; maxent_action-1 ; sbornik1 .
As a brief reminder: let we do not know probabilities (17), but we happen to known that they hold a constraint:
being realizations of some random variable. We refrain from calling energy (or minus energy), since applications of the method are general.
Now if we know precisely the average in (29), then unknown probabilities (17) can be recovered from maximizing the entropy (16) under constraint (29). In a well-defined sense this amounts to minimizing the number of assumption to be made additionally for recovering probabilities maxent_action-1 . The outcome of the maximization is well-known and was already given by us in the main text:
where the Lagrange multiplier is determined from (29).
. Importantly, the method is independent from other inference practices, though it does have relations with Bayesian statisticsskyrms-1 ; enk-1 ; cheeseman_1-1 and causal decision making skyrms-1 .
i.3.2 3.2 The open problem
But from where we could know in (29) precisely? This can happen in those (relatively rare) cases when our knowledge is based on some symmetry or a law of nature. Otherwise, we have to know from a finite number of experiments or—within subjective probability and management science buckley-1 —from an expert opinion. The former method will never provide us with a precise value of , simply because the number of experiments is finite. Opinions coming from experts do naturally have certain uncertainty, or there can be at least two slightly different expert opinions that are relevant for the decision maker. Thus in all those case we can stick to a weaker form of prior information, viz. that is known to belong to a certain interval
This problem was recognized by the founder of the method jaynes-2 , who did not offer any specific solution for it. Further studies attempted to solve the problem in several different ways:
– Following Ref. thomas-1 , which studies the entropy maximization under more general type of constraints (not just a fixed average), one can first fix by (29), calculate the maximum entropy , and then maximize over , which will mean maximizing entropy (16) under constraint (31). This produces:
– What is wrong with the simplest possibility that will state —i.e. the maximum entropy at the center of the interval—as the solution to the problem? Taking the arithmetic average of the interval independently from the underlying problem seems arbitrary.
Another issue is that each value of from the interval is mapped to the (maximum) entropy value making up an interval of entropy values. For this interval is , where and . Denoting by the inverse function of , we can take instead of .
– Ref. cheeseman_2-1 assumes that (though the precise value of the average is not known) we have a probability density for . Following obvious rules, the knowledge of translates into the joint density for maximum-entropy probabilities (30). While this is technically well-defined, it is not completely clear what is the meaning of probability density over the average . One possibility is that the random variable is sampled independently times ( is necessarily finite), and the probability density of the empiric mean
is identified with . This possibility is however problematic, since it directly relates probability of the empiric mean with the probability of the average. E.g. if we shall sample independently times from (30), then the average of the empiric mean equals , but the empiric mean itself is not distributed via .
– Given (30), one can regard
as an unknown parameter, and then apply standard statistical methods for estimating itrau-1 . Thus within this solution the maximum entropy method is not generalized: its standard outcome serves as the initial point for applying standard tools of statistics. This is against the spirit of the maximum entropy method that is meant to be an independent inference principle jaynes-2 .
– Yet another route for solving the problem was discussed in Ref. jaynes-2 . (We mention this possibility, also because it came out as the first reaction when discussing the above open problem with practioners of the maximum entropy method roger .) It amounts to the situation, where in addition to the empiric mean (36
) one also fixes the second empiric momentas the second constraint in maximizing (16). It is hoped that since identifying the sample mean (36) with the average is not sufficiently precise for a finite , then fixing the second moment will account for this lack of precision. This suggestion was not worked out in detail, but it is clear that it cannot be relevant to the question we are interested in. Indeed, its implementation will amount to fixing two different constraints, i.e. in addition to knowing precisely the average in (29), it will also fix the second moment thereby assuming more information than the precise knowledge of entails.
i.3.3 3.3 Solving the problem via bargaining
Main premises of this solution is that we should simultaneously account for both the uncertainty in and in , and that we should account for the duality of optimization, i.e. that the maximum entropy result can be also obtained via the optimization of under a fixed .
Let us first of all add an additional restriction in (31):
in the maximum entropy. We now take the joint uncertainty domain in the diagram. includes all points , where [see (37)], and where ; see (39). has the structure of the domain required by Axiom 1, with the (initial) point
being the unique anti-Pareto point, i.e. the unique point, where and jointly minimize. Recall the definition of the anti-Pareto set given in the caption of Table I.
We shall take Axiom 1 in a slightly restricted form as compared with its original form (Axiom 1) given by (6) of the main text 111The reason of this restriction is that for being able to deduce thermalization using only the restricted affine-covariance (42), we anyhow need to require the strict inequality ; cf the discussion after (8).
Axioms 2 and 4 go on as stated in the main text. In particular, Axiom 2 ensures that is a convex domain, i.e. that it does not hit the minimum entropy curve.
However, the application of axiom 3 on the affine covariance needs a restriction, because it is seen from (40) that the initial point does transform in the affine way upon affine transformations of ; cf (7). On the other hand, is obviously intact under affine transformations of that we take as the restricted form of Axiom 3 222 We emphasize that for the present problem—where we have an uncertainty interval for —the inapplicability of the affine transformation is expected for at least two reasons. Firstly, the uncertainty interval does generally change under this transformation, which indicates on altogether a different problem. Secondly, the very definition of the initial point (40) does already connect with the maximum entropy curve; hence we do not expect the full freedom with respect to affine transformations to retain. :
It should be clear from our discussion in the main text—cf. discussions after (8) and after (9)—that Axioms 1’, 2, 3’ and 4 suffice for deriving thermalization. Note that the discussion after (9) of the main text assumed only affine transformations of only, but we could consider affine transformations of only with the same success; see Fig. 3. Note as well that instead of affine transformation (42) we can apply [i.e. we can reformulate axiom (42)], if the initial state (40) is parametrized as via the inverse function of .
Axiom 5 will go as stated in the main text and demands symmetry between maximizing and maximizing . Altogether, Axioms 1’, 2, 3’, 4 and 5 suffice for deriving
as the solution of the problem with uncertainty interval (37).
i.3.4 3.4 Generalizing the solution to other types of uncertainty intervals
Uncertainty intervals that instead of (37) hold
are straightforward to deal with. Now relevant points on the maximum entropy curve can be reached by entropy maximization for a fixed , or minimizing for a fixed entropy . Hence the initial point and Axiom 1’ now read [cf. (40, 41)]:
Now we do not know a priori whether we should maximize or minimize over .
We define as above by joining together uncertainties of and , i.e. by including all points , where [see (48)], but where . The latter interval, due to (48), is where the maximum entropy values are contained. It is clear that generically has a structure of a right “triangle” formed by by two legs and a convex curve instead of the hypotenuse. Depending on whether or , we apply to either (40) with Axiom 1’ (41), or (45) with Axiom 1” (46). Axioms 2, 3’, 4, 5 apply without changes. Hence the initial point and solution reads, respectively in two regimes:
Indeed, now interpolating fromwill lead to (50), which is different as compared with interpolating (52) from . Thus we state that under (53) the prior information (48) does not suffice for drawing a unique conclusion with the bargaining method.
Fig. 5 illustrates all the above solution on an entropy-energy diagram. Two examples of the set are shown by blue, green (dashed) and red (dashed) lines. Brown dashed lines show an example of (), where the domain of states allowed according to Axiom 1 is not convex: it will cross the minimum entropy curve denoted by black lines on Fig. 5. Hence such values of are not allowed. Fig. 5 shows two examples of allowed intervals : and . The corresponding values of and are and . Blue arrows join the initial states with corresponding Nash solutions (43, 47).
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