What happens when strategies reproduce instead of reasoning?
Topic 23 keeps one decision-maker but gives them several objectives that genuinely conflict, so “optimal” stops being a point and becomes a curve. Game theory (topics 11 to 15) drops the other assumption: when another party is also choosing in response to you, your best plan depends on theirs and there is no single answer to compute, only an equilibrium to find. It underwrites markets that clear through auctions, security proofs written as games between an adversary and a defender, and populations that settle into stable mixes of strategy.
Every topic above assumes a rational player choosing a strategy by reasoning about payoffs. Evolutionary game theory removes the reasoning entirely. Strategies are not chosen; they are inherited or copied, and whichever strategy currently earns more simply becomes more common next round: through literal reproduction in biology, or through imitation, learning and market share anywhere else. The question shifts from "what should a rational player do?" to "what mix of strategies, once reached, resists being invaded?"
The classic setting is the Hawk–Dove game (Maynard Smith & Price, 1973): contest a resource aggressively (Hawk) or back down (Dove). All-Hawk is not stable: constant injury costs more than the resource is worth, so a rare Dove, who never fights, does better than average and spreads. All-Dove is not stable either: a single Hawk walks in and wins every contest uncontested. The population is pulled toward a specific mixture from both directions, which is precisely what "evolutionarily stable" means: a strategy mix that, once established, cannot be invaded by any rare alternative.
The bridge back into this course is direct: genetic algorithms (topic 10) are this idea, deliberately engineered. Breeding a population of candidate solutions and keeping the fitter half is evolutionary game dynamics run on purpose as a search method, rather than observed in nature.
Payoffs for the Hawk–Dove game, resource value V, cost of injury C, with C > V (losing a fight costs more than the resource is worth, the interesting case):
Hawk vs Hawk: (V − C) / 2 each (fight; winner takes V, loser pays C, 50/50)
Hawk vs Dove: Hawk gets V, Dove gets 0
Dove vs Dove: V / 2 each
Let p be the fraction of the population playing Hawk. The evolutionarily stable strategy (ESS) is the p where Hawk's average payoff exactly equals Dove's: the crossing point in the figure, where neither strategy has an edge and the mix stops changing:
payoff(Hawk) = p·(V−C)/2 + (1−p)·V = payoff(Dove) = p·0 + (1−p)·V/2
solving: p* = V / C
Worked with V = 2, C = 4 (a fight costs twice what the resource is worth): p* = 2/4 = 0.5, matching the figure exactly. Check by substitution: payoff(Hawk) at p=0.5 is 0.5·(−1) + 0.5·2 = 0.5; payoff(Dove) is 0.5·0 + 0.5·1 = 0.5. Equal, confirming the crossing point. If C < V instead, fighting is worth it outright and all-Hawk becomes the stable strategy: the mixed equilibrium only exists because injury is assumed to cost more than the prize.
The population-level version of "strategies that do better spread" is the replicator equation: ẋᵢ = xᵢ · (fᵢ(x) − f̄(x)), where xᵢ is strategy i's population share, fᵢ its current payoff, and f̄ the population's average payoff. A strategy grows exactly when it beats the average: the continuous-time engine that pulls the population toward an ESS from either side, as drawn above.
Treating cancer as an evolving population Standard chemotherapy hits a tumour at maximum tolerated dose, which kills drug-sensitive cells fast and (precisely because it removes their competition) hands the field to drug-resistant cells, which then take over and the treatment fails. Adaptive therapy treats the tumour as competing strategies in an evolutionary game instead: dose just enough to control the sensitive population while deliberately leaving it around to keep out-competing the resistant cells for space and resources, rather than trying to eliminate everything at once. Zhang, Cunningham, Brown & Gatenby, "Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer," Nature Communications 8 (2017): in a real clinical trial, standard continuous dosing drove rapid selection for resistant cells and treatment failure, while adaptive, evolution-aware dosing suppressed the resistant population's growth for substantially longer. The mathematics is the Hawk–Dove machinery above, with tumour cell lines standing in for Hawks and Doves.
Check yourself
In the Hawk-Dove game with V = 2 and C = 4, what fraction of the population plays Hawk at the evolutionarily stable mix?
The stable mix is where Hawk and Dove earn the same: p = V/C = 2/4 = 0.5. Check: at p = 0.5 both strategies earn 0.5.