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This chapter explores the concept of mixed policies and how the notions for pure policies can be adapted to this more general type of policies. A pure policy consists of choices of particular actions (perhaps based on some observation), whereas a mixed policy involves choosing a probability distribution to select actions (perhaps as a function of observations). The idea behind mixed policies is that the players select their actions randomly according to a previously selected probability distribution. The chapter first considers the rock-paper-scissors game as an example of mixed policy before discussing mixed action spaces, mixed security policy and saddle-point equilibrium, mixed saddle-point equilibrium vs. average security levels, and general zero-sum games. It concludes with practice exercises with corresponding solutions and an additional exercise.

This chapter takes a first look at strategic interdependence, known to biologists as frequency dependent selection. It distinguishes positive interaction effects (increasing returns or synergy) from negative effects (e.g., congestion). The chapter introduces Maynard Smith’s famous Hawk‐Dove game, and goes on to show that there are only two other generic types of 2 × 2 games: dominant strategy (DS), illustrated with biological data on RNA viruses and economic examples drawn from eBay; and coordination (CO) games. Conditions for the prisoner’s dilemma are defined. The remaining sections show how these games represent edges of the state space in games with three alternative strategies. This includes data on male side‐blotched lizards that shows they play the true rock‐paper‐scissors: each edge game has a different dominant strategy, forming an intransitive loop around the 2‐dimensional simplex.

This chapter analyzes more complex forms of interaction. Using the rock-paper-scissors game of the European common lizard as an example, it introduces helpful geometric techniques for analyzing 3 × 3 and higher dimensional games and shows how to extend them to nonlinear games. The authors then consider two-population games, both the usual “asymmetric” analysis and own-population effects. The chapter also introduces more general dynamics than replicator, including several forms of monotone dynamics. Finally, it explains how to estimate the payoff matrix from time-series data on the side-blotched lizard using maximum likelihood techniques and state-of-the-art multinomial logit and Dirichlet models.

This chapter formalizes spatial strategic interaction as games on grids. It develops cellular automata examples ranging from Conway’s classic Game of Life to complex cooperation, and revisits several earlier examples of assortative and disassortative interactions in an explicit spatial framework. The chapter features simulations of viscous game dynamics with emergent features such as cooperative groups in public goods games, including density and frequency dependent interactions. Two-population versions of the cellular automata are developed (e.g., for the buyer-seller game). Esherichia coli bacteria provide an example of a spatial rock-paper-scissors game: a colicin-producing/colicin-resistant type is beaten by a colicin-resistant type, which is beaten by an undefended type, which is beaten in turn by the first type.

This chapter documents general RPS interactions in mating systems for isopods, damselflies, fish, and birds. It applies Hadamard products to show how mate preferences can change one RPS game into another. Regular RPS cycles in one sex favor invasion of a rule of thumb in the other sex in which common mating types are avoided and rare types are preferred. From such interactions, the one-population true RPS game is converted into an apostatic RPS game. The RPS can also be broken if mates evolve preference for self-mating types, with implications for cooperation and speciation. A new co-evolutionary model describes predator/prey interactions with learning. The model is referred to as ABC-NR after the conspicuous and toxic aposematic model, harmless Batesian mimics, and cryptic types in prey, while predators are either naÏve or responsive to aposematic signals. Fisher’s conjecture of an advantage from prey clustering is required for an interior ESS.