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Dai Zusai

Gains in evolutionary dynamics A unifying and intuitive approach to linking static and dynamic stability. Dai Zusai. Philadelphia, U.S.A. DZ’s research agenda. Stability of equilibrium in evolutionary dynamics. Equilibrium in a market. market equilibrium. Supply. Demand.

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Dai Zusai

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  1. Gains in evolutionary dynamicsA unifying and intuitive approach to linking static and dynamic stability • Dai Zusai Philadelphia, U.S.A.

  2. DZ’s research agenda Stability of equilibrium in evolutionary dynamics

  3. Equilibrium in a market market equilibrium Supply Demand

  4. Equilibrium in a (population) game "Payoff" of each strategy Strategy A Strategy B : Share of each strategy in the population

  5. Equilibrium, expressed in a model 0 0 0 …... Market model: ’s: supply−demand of each good z’s: price of each good Game theory: ’s: relative payoff of each strategy z’s: share of each strategy

  6. DZ’s research agenda Stability of equilibrium in evolutionary dynamics

  7. Evolutionary dynamics: construction Game Incentives Choices Payoffs of strategies Shares of strategies Evolutionary dynamics Individual agent’s decision of switching the choice Aggregation of individual agents’ switches Individual agent’s switching rate An agent occasionally reconsider the choice, when it receives a “revision opportunity,” which arrives randomly. (To make the dynamic differentiable with respect to infinitesimal change in time.) At that opportunity, the agent finds the candidate of a new strategy, and decides whether or not to switch to it. At each moment of time, for each strategy, we count the agents who switches to it and those who switches from it. Sandbox set-up

  8. Evolutionary dynamics: variety Evolutionary dynamics Decision rule Find the optimal strategy (simply, greatest payoff) among all the strategies. Switch to it, regardless of the amount of the payoff improvement. Best Response Dynamic Find the optimal strategy (simply, greatest payoff) among all the strategies. Switch to it with a probability proportional to the payoff difference from my current payoff. Tempered Best Response Dynamic Decision rules in Evolutionary dynamics Possibly, not exactly optimizing. Possibly, reluctant to switch. Sample another strategy randomly.Switch to it with a probability proportional to the payoff difference from the average payoff. Excess Payoff Dynamic Sample another strategy randomly.If it performs better than my current strategy, switch to it with a probability proportional to the payoff difference from my current payoff. Pairwise comparison dynamics Sample another agent randomly and observe the agent’s strategy.If it performs better than my current strategy, switch to it with a probability proportional to the payoff difference from my current payoff. Imitative dynamics

  9. Applications

  10. DZ’s research agenda Stability of equilibrium in evolutionary dynamics

  11. Stability

  12. John Hicks Equilibrium stability in economic theory proposed the notion of static stability, based on economic intuition on incentives and rational choices. Value and Capital, 1939 (Ed.2 1946) Paul Samuelson started rigorous investigation of dynamic stability, based on mathematical theory of differential equations. 3 papers in Econometrica 1941, ’42, ‘44

  13. Static stability in market models What if the price were $2.00, higher than the equilibrium price? At a price of $2.00, there is a negativeexcess demand (“surplus”) of 2 million bottles. Then, it is natural to expect the price to go down thanks to the market mechanism. Negative correlation between changes in excess demand and changes in prices Source: Hubbard & O’Brien, Microeconomics

  14. Static stability in a game "Payoff" of each strategy Strategy A Strategy B : Share of each strategy in the population

  15. Static stability in a game Global/local stability "Payoff" of each strategy Strategy A Strategy B Negative correlation between changes in shares and changes in payoffs Strategy A : Share of each strategy in the population

  16. Static stability Economically natural adjustments of prices Game Incentives Choices Prices of goods Excess demand ofgoods Hicksian stability Excess demand • Economically reasonable updates of choices Incentives Choices Evolutionary stability (ESS) Share ofstrategies Payoff ofstrategies

  17. Static stability Negative correlation between change in choices and change in incentives Partial equilibrium (There’s only one option; Binary choice of taking it or not) General equilibrium (There’s many options; Choice from them) Market Law of supply/demand Hicksian stability Evolutionary stability Game Negative externality

  18. Mathematical completion Price adjustment dynamic Economically natural adjustments of prices Game Incentives Choices Prices of goods Excess demand ofgoods Combined dynamic Excess demand • Evolutionary dynamic • Economically reasonable updates of choices Incentives Choices Combined dynamic Share ofstrategies Payoff ofstrategies

  19. (Mathematically true) dynamic stability Obtain the combined dynamic as an autonomous differential equation: Check whether as using some mathematical conditions about stability of a solution of a differential equation.

  20. Static stability Dynamic stability even if the dynamic seems to be compatible with incentives. e.g. Friedman (91, Ecta) order compatibility: : a linear payoff function (random matching in a normal form game) with a unique interior ESS a stable game : represents a range of feasible transition vectors under order compatibility —Take the most outward one at each state Escaping from the ESS. e.g. Milchtaich, Hofbauer & Sandholm

  21. Static stability Dynamic stability

  22. Discontent to each other Incomplete Too loose Groundless Nonsense Too specific No intuition

  23. Abstract: The goal of this paper is to present a universal and intuitive approach to link static stability with dynamic stability, i.e., asymptotic stability of equilibrium, through economic principles behind evolutionary dynamics. In economics, static stability is traditionally defined as a notion about how incentives respond to choices: for a population game, it means that, on average, the payoff of a strategy responds negatively to an increase in the share of the strategy. This is characterized by negative definiteness of the Jacobian matrix of the payoff function and satisfied globally in a contractive game and locally at a regular ESS. We say an evolutionary dynamic is economically reasonable, if an agent’s choice of a new strategy can be justified as an optimal choice possibly by introducing additional costs and constraints; for example, inertia can be justified by switching costs, and switches to suboptimal strategies can be by restriction to available strategies. This class of dynamics includes not only the best response dynamic, pairwise payoff comparison dynamics and excess payoff dynamics. The key in our proof is a net gain from revision of the strategy, i.e., the payoff improvement by the revision minus the switching cost. Static stability implies that the aggregate net gain monotonically decreases over time under an economic reasonable dynamic and thus can be used as a Lyapunov function. Our Lyapunov function allows us to extend the dynamic stability to mixture of heterogeneous populations, who may follow different dynamics or different payoff functions. While our analysis here is confined to normal form games for clear illustration of our approach, we argue that our approach is promising for further applications to more complex situations.

  24. What did I do? • Proved that the economists’ static stability condition (in a game) is MATHEMATICALLY RIGHT as a sufficient condition for dynamic stability! • If the condition is satisfied, equilibrium is stable under any evolutionary dynamic as long as the underlying decision rule is solely based on “economic reasoning.” • The theorem encompasses a general class of evolutionary dynamics that based on “economic reasoning.” • The stability holds even if we cannot numerically identify the dynamic, we mix different dynamics, or we run simulations of finite-agent models. • Of course, it is mathematically rigorously proven based on differential equations (with slightly new, purely mathematical theorems). • The idea of the proof comes from economic intuition, which can be generally applicable to different settings.

  25. Gravity and height As long as the ball is driven by gravity, the height of the ball must go down(never up); when the height is at the bottom, the ball must be in either one dish.

  26. Appendix Basic set up as a sandbox Basic set-up: very basic as in Sandholm’s book (2010) Game :a single population game in a normal form • Continuously many agents in a single population • An agent chooses an action (strategy) from a finite set of actions The action distribution determines payoffs as . Evolutionary dynamic : a deterministic dynamic, constructed from a revision protocol • Each agent receives a revision opportunity from a Poisson process (arrival rate=1). • Observing the current payoff vector , the agent decides whether or not to switch the action and, if so, which action to take, following a revision protocol. Aggregation of individual agents’ revision processes yields a deterministic dynamic of s.t. . Game dynamic : an autonomous dynamic of generated from combination of and . Construction

  27. Static stability in a game: not hold "Payoff" of each strategy Strategy A Strategy B : Share of each strategy in the population

  28. Static stability in a game "Payoff" of each strategy Strategy B Strategy A : Share of each strategy in the population

  29. Static stability in a game: multiple "Payoff" of each strategy Strategy A Strategy B : Share of each strategy in the population

  30. Static stability Known sufficient conditions on for dynamic stability over major dynamics: • Global stability: is a stable (contractive) game(HS ‘09), i.e., for any. • Local stability: is a regular ESS (Taylor&Jonker ‘78, Friedman ‘91, Sandholm ‘10), i.e., if and as long as if . at any in a neighborhood of . Static stability: Negative semi-definiteness of Hessian Economic interpretation: payoffs respond negatively to increases in shares. (HS ‘09 “Self-defeating externalities”) • Cf. Milchtaich (‘19) “Static stability of equilibrium” attempts to unify the existing equilibrium notions in various settings (ESS, CSS, etc.) --- “static stability” is defined by assuming a linear returning path and calculating the overall payoff improvement from the return. Stat stability in games

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