The universal payoff is not money, utility, or fitness. It is entropy. Game theory works because behavior itself follows the second law of thermodynamics, consuming free energy along the paths of least action.
Von Neumann founded game theory on thermodynamics and Nash borrowed his equilibrium from chemistry. This paper takes the resemblance seriously and finishes the mapping.
All games, from molecules to markets to nations, are the same process at different scales: flows of energy consuming free energy as fast as possible.
Entropy increase in the least time is the universal payoff.
Von Neumann compared every exchangeable asset to an energy density, kBT per entity, then set the approximation aside and used money as payoff because he could not do better.
Nash took his equilibrium concept from Gibbs' chemical equilibrium but never made the variables one-to-one with energy.
This paper completes the bridge using statistical physics of open systems.
The reason no single payoff function ever worked is that utility, money, fitness and information are different words for the same thing: energy differentials.
| game theory | physics |
|---|---|
| payoff function | rate of entropy increase |
| utility / money / fitness | free energy |
| assets in possession | energy densities |
| strategy / move | mechanism channeling a flow |
| Nash / ESS equilibrium | free-energy minimum |
| zero-sum game | closed Hamiltonian system |
| non-zero-sum game | open system with energy influx |
| a piece of information | an asset bound to energy |
Boltzmann counted isoenergetic microstates, like pips on dice. That probability is fixed and describes closed, stationary systems. The paper instead uses a probability that evolves with energy, the key to open, evolutionary games.
Probability counts the equivalent configurations of a conserved system. It is constant in energy, so it can only describe re-shuffling among states that already exist. This is classical statistical mechanics, and it is why it stops short of living, economic, or evolutionary systems.
Probability P here is conditional on energetic state and changes as energy flows in or out. It describes non-conserved systems that evolve. This is the foundation the rest of the paper stands on: the equation of motion for an open system is the principle of increasing entropy.
Equation 1 is the heart of the paper. Every transaction produces a payoff, and because each term is a square, the universal payoff can never go down.
dtS is the rate of entropy increase, the payoff.
Ajk is the free-energy driving force of a transaction between assets j and k.
đNj is the change in holdings of asset j caused by the move.
Δμjk = potential (value) difference between the assets.
iΔQjk = energy influx from the surroundings (income into an open game).
Substituting the equation of motion below yields dtS ∝ Σ Ajk², so entropy rises almost everywhere.
The rate at which a move changes your holdings is proportional to the driving force Ajk, scaled by σjk, the efficiency of the mechanism channeling it. A better trading channel, tool, or strategy is simply a larger σjk.
Plug Equation 2 into Equation 1 and every entropy term becomes σ · A². Because a square is never negative and σjk is positive, entropy increases almost everywhere. The second law is not postulated, it drops out of the algebra of a flow following its own driving force.
A buyer values a good at 9 energy units, the seller at 4. The free-energy gap is Ajk = Δμ = 9 − 4 = 5.
Same deal, same gap. The sharper mechanism drains the free energy three times faster and produces three times the entropy. This is why σjk matters: it is the whole advantage of a better tool, market, or strategy, expressed as one number. Neither payoff can be negative.
Systems do not minimize energy, they minimize time. Flows follow geodesics, the optimal paths that drain free energy soonest. This is the maximum entropy production principle, identical to the principle of least action.
The product form of Equation 1 ties every move to the holdings it changes. With three or more degrees of freedom the system is non-Hamiltonian: each move reshapes the motives for every future move. Drag the slider to feel the threshold.
A move changes the assets, and the assets are the driving force for the next move. Behavior and motive are inseparable.
This is exactly the interdependence between your strategy si and everyone else's s-i. Decisions depend on decisions that depend back on the first decision.
The intractability of extensive-form games is not a computing limit. It is an inherent property of open systems with three or more degrees of freedom.
For a large, near-equilibrium system, S = kB ln P is a smooth, sufficient statistic and you can extrapolate. But when a single move carries energy on the order of the whole system (Ajk ≈ kBT), the approximation fails and probability moves in discrete steps.
Example. A founder's startup is acquired. One transaction re-orders their entire status Pj; it is not a smooth slide, it is a step. You cannot extrapolate the next move from the last, because the move itself re-shaped the landscape. This long-tail event is exactly where prediction fails and where the chaos above becomes visible.
When every driving force vanishes, Ajk=0, no move improves anyone's lot. This free-energy minimum is Lyapunov-stable, and the asset partition is not flat. It is log-normal.
The equilibrium partition of assets. Not a design choice, a consequence of the least-time principle.
The equilibrium holds against variation within existing strategies, but a new strategy that taps previously unreachable resources perturbs it. The system finds a deeper minimum. This is biological speciation, market disruption, and bifurcation, the same event at different scales.
Entropy is the logarithmic status of the whole game over the product of each player's probability Pj. It climbs toward Smax as the game plays out.
Payoff hits zero only when every driving force vanishes. No move by any player can improve their lot. This is the reaction-equilibrium condition, and it forces the skewed partition shown beside it.
Example. Picture a used-car market. Arbitrageurs move cars until no dealer can profitably relocate one: every value gap Δμjk has closed, so Ajk = 0 for all pairs. What remains is not an even split of inventory. It is skewed: a few dealers hold most of the cars, most hold a few, many hold none. The same log-normal curve fits personal wealth, species abundance in an ecosystem, and molecules partitioning among energy levels. Different games, one equilibrium shape.
Cooperation is not an exception to competition. It is a strategy for it. A group accesses more resources and channels more flow than its members could alone, so it reaches a higher entropic status.
Every form of information is bound to a physical representation, so it is subject to the laws of thermodynamics. The flow of energy is the only means of conveying information, which makes them the same currency.
Learning is just the forming of new geodesics for the flows of energy that carry information.
Acquiring information is acquiring assets that open new paths to reduce free energy.
No observation without interaction. An observer always perturbs the energy of the game and integrates into its course. There is no outside view.
Information has a price. Gathering it costs energy, so players stop searching when the expected gain falls below the cost. This is rational ignorance.
"Rationality" is often mimicry. High-energy players impose their norms on others. Much of what looks like rational consensus is submission to the flows that dissipate fastest.
Decision-making is subjective because payoff depends on your holdings. Mimicry succeeds because pioneers blaze suboptimal but real trails. The tragedy of the commons is the 2nd law draining a society whose surroundings hold less energy than it does.
There is no universal rational choice. Payoff is a function of your possessions and moves. What looks irrational to an observer may be optimal given that player's holdings and the future they discount.
Early on, almost any move produces entropy, so pioneers explore by trial and error. Successors imitate because following a blazed trail beats the cost of search. This is why suboptimal standards get cemented.
When exploitation drops the energy in the surroundings below the energy in the social system, the 2nd law reverses the flow: energy drains out of society, exactly when it needs cohesion most.
This paper is the thermodynamics-to-economics bridge that the Synchronicity platform (2021) leans on to formalize money as energy and incentives as physical gradients.