Project
AA3-18
Evolution Processes for Populations and Economic Agents
Project Heads
Max Klimm, Maite Wilke Berenguer
Project Members
Tobias Paul (HU)
Project Duration
01.04.2024 – 31.03.2026
Located at
HU Berlin and TU Berlin
Description
This project aims to connect the research areas of evolutionary game theory and population genetics. Particularly with regards to the learning model of adaptive play there are some striking similarities but also subtle differences between evolutionary game theory and reproduction mechanisms in Wright-Fisher models of population genetics. This will lead to new learning models for agents in competitive environments and new mathematical methods to study their behavior.
In adaptive play, each player of the game playing in round n are allowed to select from the past h rounds of plays of previous players a total of s ≤ h rounds. Looking at the empirical distribution of the opponent’s plays in the past, the current player will play the best response to this empirical distribution. For example, in the simple case of the coordination game with two players and two choices as well as s = 1 and arbitrary h ≥ 1, each player samples a previous play of the opposing player for which the best reply is to copy the observed play.
In the Wright-Fisher model, there are h individuals alive in each generation and the next generation is determined by multinomial resampling of the previous generation. This resampling mechanism copies the types of the parent individuals to the offspring. In a variant of this process, one can allow for slower evolution by only allowing exactly one individual to produce exactly one additional offspring. This then is reminiscent of the adaptive play mechanism above where the types correspond to plays in the past and the single reproduction reflects the copy of one previous strategy.
A subtle difference comes to light when considering that in adaptive play, one can only choose from the past h games with h fixed. In the one-step Wright-Fisher setup however, the types can remain available for longer than h generations just by chance. In order to employ the tools of population genetics we thus modify the adaptive play mechanism to not consider the past h plays for some fixed h, but instead we create a reservoir of h plays which a player can choose from and in each round of play the previous play of the opponent replaces one of the plays in the reservoir uniformly at random. Then, a given play remains available for a number of rounds of play given by a geometric distribution with parameter 1/h. We then track the frequency of one of the two available choices in each of the players reservoir.
We expect a separation of time scales to occur in this version of adaptive play: On a fast time scale we should see that the observed frequencies of plays in the reservoir for each player to balance out. Then, on a slower time scale, we anticipate stochastic fluctuations of the frequency process according to a Wright-Fisher diffusion which is identical for both player reservoirs (see example simulation below).
Project Webpages
Selected Publications
Selected Pictures
A simulation of the frequencies with h=10^4 for a time of up to 10*10^4 plays started with frequencies 0.2 and 0.7. One can see the deterministic averaging of the frequencies in the beginning.
A simulation of the frequencies with h=10^4 for a time of up to 10^8 plays started with frequencies 0.2 and 0.7. One can see the stochastic nature of the frequencies until eventually we are left with a frequency of 1 in an absorbing state.
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