Fig 1.
A schematic representation of behavioural mistakes in a rock-paper-scissors game.
(A) Rock-paper-scissors dynamics with pure strategies is described by a fitness matrix such that the cyclic relationship between the three strategies is promoted. (B) The effect of execution errors on the example of one interaction: here individual 1 has chosen strategy paper and individual 2 has chosen strategy rock. Without mistakes, individual 1 would win this instance of the contest. However, a mistake in the execution leads to mixed strategies being played for both individuals resulting in different possible outcomes of the interaction. Hence, the outcome of the game is no longer deterministic but stochastic and depends on the probability distribution of mistakes.
Fig 2.
Game flow for the unstable RPS game with uniform mixed strategies for different values of λ.
Here, a stable fixed point is denoted by a red circle and a unstable fixed point is denoted by a white circle. The colour in the interior of the simplex indicates the rate of change: from slow (blue) to fast (red). In this example, completely random execution errors lead to the dominance of the rock strategy. We use the Wolfram Mathematica project [49] to produce these phase planes.
Fig 3.
Game transitions under execution errors from example 2.
(A) Frequencies of each strategies in the interior equilibrium as functions of λ. Here, x1 represents rock frequency, x2—paper frequency and x3—scissors frequency. The interior equilibrium exists for most the values of λ but (,
). Further, the coloured bar at the top of the plot indicates stability intervals of λ for different vertices (a stable vertex is indicated on top of the bar). For instance, vertex 3 is the only stable vertex for λ ∈ (≈ 0.287,
). Game flow for the unstable rock-paper-scissors game is depicted for different values of λ as follows: (B) λ = 1, (C) λ = 0.3, (D) λ = 0.27, (E) λ = 0.22, (F) λ = 0.205, (G) λ = 0.16, (H) λ = 0.1, (I) λ = 0.05, (J) λ = 0. We depicted each transition in the game from panel A. Here, a stable fixed point is denoted by a red circle and a unstable fixed point is denoted by a white circle. Hence, as λ changes its values from 1 to 0, the game experiences several transitions in its equilibria and for different degrees of execution errors, each of the pure strategies has a chance to dominate. However, for the maximum plasticity, only pure rock strategy survives.
Fig 4.
Various game transitions of the unstable RPS game as λ varies between [0, 1].
The components of the interior fixed point are plotted as functions of λ. Further, the coloured bar at the top of the plot indicates stability intervals of λ for different vertices (a stable vertex is indicated on top of the bar). (A) The interior fixed point exists for all λ but vertices interchange their stability. In the limit of mistakes (λ → 0), two vertices are stable. (B) The interior fixed point exists for a sub-interval and vertices interchange their stability. As λ → 0, two vertices are stable. (C) The interior fixed point exists for two sub-intervals of (0, 1). In the limit of mistakes (λ → 0), only vertex 1 is stable. (D) The interior fixed point exists for almost all values of λ. In the limit of mistakes (λ → 0), all three vertices are stable. Generally, the exact equilibria transitions and existence of an interior equilibrium is determined by the limiting distribution of mistakes, S. We found that for almost all matrices S there is a high chance that at least one of the pure strategies will become dominant.
Fig 5.
A schematic representation of a possible influence of execution errors on the RPS dynamics.
The original game possesses an unstable equilibrium . Once the execution errors are introduced, for some λ the game can obtain a stable equilibrium
represented by a vertex i. As the probability to play mixed strategies in this case is high, it keeps disturbing the strategy execution of the players choosing strategy i according to Eq (7) resulting in an equilibrium
that is possibly in the interior.