Applications of Game Theory
to Evolutionary Biology
Mixed-strategy solutions for
2x2 games can be calculated
arithmetically. More complicated games with multiple
strategies require the use of matrix algebra: an
entertaining and enlightening book is JD Williams' "The
Compleat Strategyst" (2nd ed., 1966).
In the so-called "Bomber Defense Problem," Blue has two bombers, one of which carries the bomb and the other of which carries equipment for defense & counter-measures, etc. The bombers fly in in a formation that provides one with protection by the other: the bomb carrier can fly in the unprotected (Blue 1) or protected (Blue 2) position. Red has a single fighter, which can attack either the unprotected (Red 1) or protected bomber (Red 2), but not both. This particular example was featured prominently by the RAND Corporation in initial discussions of Game Theory as applied to military situations.
Blue wins if the bomb carrier gets through. Convert the probabilities of bomber success to payoffs, with p = 100% = 5). This will happen if the Red fighter attacks the wrong bomber no matter where it is (prob 100%: payoff 5), or may happen if the Blue bomber is attacked in the unprotected (prob 60%: payoff 3) or protected (prob 80%: payoff 4). The value of the game to Blue is 4.33. Compare this to the payoff of 4, if both sides do what comes naturally, which is for Blue always to place the bomb-carrier in the protected position (Blue 2), and for Red always to attack that position (Red 2). Note that the payoff to Blue is the same if Blue uses the mixed strategy and Red always attacks the protected bomber, not knowing what strategy Blue has adopted. The mixed strategy produces a minimax payoff. The Value to Blue in adopting a mixed strategy is (4.33 - 4.00) / 4.00 = 8.25%, which is a marked gain.
The "Attack-Defense problem" is applicable to within-species interactions subject to Natural Selection. For example, various species defend territories that include several resource types of differing values. Interference Competition occurs when conspecifics on adjacent territories actively interfere with their neighbors in an attempt to gain resources. Blue has nesting and feeding areas (the latter more valuable in non-breeding seasons), and can defend one or the other (Blue 1 & Blue 2) but not both. Red may attack either area (Red 1 & Red 2), but not both. Blue can defend successfully an attack on either area (payoff 3). A Red attack on an undefended nesting area is a minor loss to Blue (payoff -1), whereas a successful attack on an undefended feeding area is a major loss (payoff -9). The Value of a 3:1 mixed strategy for Blue is a break-even 0, compared with -1 for consistent defense of the more valuable resource at the cost of the less valuable: the gain to Blue is (3 - [-1]) / 3.00 = 1.33%, a modest gain. Note that this game is a 'zero-sum' game around an equilibrium, because gains and losses to the two players cancel.
If the payoff for a within-species interaction is a component of evolutionary Fitness, Game Theory predicts evolution of flexible optimal attack-defense behaviors, noting that single strategies are dis-favored by Natural Selection. Note as well that the relative values of the nesting and feeding areas will change seasonally, introducing another complexity into the game. Finally, games may become non-zero-sum games if the competing conspecifics place different values on resource areas: a male may place zero value on a nesting area, and a value of 10 on a feeding area.
In the so-called "Bomber Defense Problem," Blue has two bombers, one of which carries the bomb and the other of which carries equipment for defense & counter-measures, etc. The bombers fly in in a formation that provides one with protection by the other: the bomb carrier can fly in the unprotected (Blue 1) or protected (Blue 2) position. Red has a single fighter, which can attack either the unprotected (Red 1) or protected bomber (Red 2), but not both. This particular example was featured prominently by the RAND Corporation in initial discussions of Game Theory as applied to military situations.
Blue wins if the bomb carrier gets through. Convert the probabilities of bomber success to payoffs, with p = 100% = 5). This will happen if the Red fighter attacks the wrong bomber no matter where it is (prob 100%: payoff 5), or may happen if the Blue bomber is attacked in the unprotected (prob 60%: payoff 3) or protected (prob 80%: payoff 4). The value of the game to Blue is 4.33. Compare this to the payoff of 4, if both sides do what comes naturally, which is for Blue always to place the bomb-carrier in the protected position (Blue 2), and for Red always to attack that position (Red 2). Note that the payoff to Blue is the same if Blue uses the mixed strategy and Red always attacks the protected bomber, not knowing what strategy Blue has adopted. The mixed strategy produces a minimax payoff. The Value to Blue in adopting a mixed strategy is (4.33 - 4.00) / 4.00 = 8.25%, which is a marked gain.
The "Attack-Defense problem" is applicable to within-species interactions subject to Natural Selection. For example, various species defend territories that include several resource types of differing values. Interference Competition occurs when conspecifics on adjacent territories actively interfere with their neighbors in an attempt to gain resources. Blue has nesting and feeding areas (the latter more valuable in non-breeding seasons), and can defend one or the other (Blue 1 & Blue 2) but not both. Red may attack either area (Red 1 & Red 2), but not both. Blue can defend successfully an attack on either area (payoff 3). A Red attack on an undefended nesting area is a minor loss to Blue (payoff -1), whereas a successful attack on an undefended feeding area is a major loss (payoff -9). The Value of a 3:1 mixed strategy for Blue is a break-even 0, compared with -1 for consistent defense of the more valuable resource at the cost of the less valuable: the gain to Blue is (3 - [-1]) / 3.00 = 1.33%, a modest gain. Note that this game is a 'zero-sum' game around an equilibrium, because gains and losses to the two players cancel.
If the payoff for a within-species interaction is a component of evolutionary Fitness, Game Theory predicts evolution of flexible optimal attack-defense behaviors, noting that single strategies are dis-favored by Natural Selection. Note as well that the relative values of the nesting and feeding areas will change seasonally, introducing another complexity into the game. Finally, games may become non-zero-sum games if the competing conspecifics place different values on resource areas: a male may place zero value on a nesting area, and a value of 10 on a feeding area.
Examples after JD Williams (1966); Figure & Text material © 2021 by Steven M. Carr