Bimatrix games that include interaction times alter the evolutionary outcome: The Owner-Intruder game.

Classic bimatrix games, that are based on pair-wise interactions between two opponents in two different roles, do not consider the effect that interaction duration has on payoffs. However, interactions between different strategies often take different amounts of time. In this article, we further develop a new approach to an old idea that opportunity costs lost while engaged in an interaction affect individual fitness. We consider two scenarios: (i) individuals pair instantaneously so that there are no searchers, and (ii) searching for a partner takes positive time and populations consist of a mixture of singles and pairs. We describe pair dynamics and calculate fitnesses of each strategy for a two-strategy bimatrix game that includes interaction times. Assuming that distribution of pairs (and singles) evolves on a faster time scale than evolutionary dynamics described by the replicator equation, we analyze the Nash equilibria (NE) of the time-constrained game. This general approach is then applied to the Owner-Intruder bimatrix game where the two strategies are Hawk and Dove in both roles. While the classic Owner-Intruder game has at most one interior NE and it is unstable with respect to replicator dynamics, differences in pair duration change this prediction in that up to four interior NE may exist with their stability depending on whether pairing is instantaneous or not. The classic game has either one (all Hawk) or two ((Hawk,Dove) and (Dove,Hawk)) stable boundary NE. When interaction times are included, other combinations of stable boundary NE are possible. For example, (Dove,Dove), (Dove,Hawk), or (Hawk,Dove) can be the unique (stable) NE if interaction time between two Doves is short compared to some other interactions involving Doves.

Classic bimatrix games, that are based on pair-wise interactions between two opponents in two different roles, do not consider the effect that interaction duration has on payoffs. However, interactions between different strategies often take different amounts of time. In this article, we further develop a new approach to an old idea that opportunity costs lost while engaged in an interaction affect individual fitness. We consider two scenarios: (i) individuals pair instantaneously so that there are no searchers, and (ii) searching for a partner takes positive time and populations consist of a mixture of singles and pairs. We describe pair dynamics and calculate fitnesses of each strategy for a two-strategy bimatrix game that includes interaction times. Assuming that distribution of pairs (and singles) evolves on a faster time scale than evolutionary dynamics described by the replicator equation, we analyze the Nash equilibria (NE) of the time-constrained game. This general approach is then applied to the Owner-Intruder bimatrix game where the two strategies are Hawk and Dove in both roles. While the classic Owner-Intruder game has at most one interior NE and it is unstable with respect to replicator dynamics, differences in pair duration change this prediction in that up to four interior NE may exist with their stability depending on whether pairing is instantaneous or not. The classic game has either one (all Hawk) or two ((Hawk,Dove) and (Dove,Hawk)) stable boundary NE. When interaction times are included, other combinations of stable boundary NE are possible. For example, (Dove,Dove), (Dove,Hawk), or (Hawk,Dove) can be the unique (stable) NE if interaction time between two Doves is short compared to some other interactions involving Doves. π e 11 , π f 11 π e 12 , π f 12 e 2 π e 21 , π f 21 π e 22 , π f

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(1) where π e ij (respectively, π f ij ) is the payoff to e i (respectively f j ) when interact-102 ing with f j (respectively e i ). In contrast to classic evolutionary game theory, we explicitly incorporate the duration of interactions into the game through 104 the time interaction matrix 105 f 1 f 2 e 1 τ 11 τ 12 e 2 τ 21 τ 22 (2) where τ ij is the expected time two players using strategy e i and f j stay 106 together.
The corresponding time-constrained bimatrix game based on payoff bimatrix

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(1) and time interaction matrix (2) is then the two-strategy game with payoffs 144 given by the fitness functions (9)  .
(10) 1 We will use the phrase "fitness functions" rather than "payoffs" for these timeconstrained games from now on to avoid confusion with payoffs in (1).
2 If (e 1 , f 1 ) is a strict NE, it must also resist invasion by mutants in population 1 that use any other strategy (including a mixed strategy) besides e 1 . However, since the fitness of the focal mutant is linear in the components of its mixed strategy, it is enough to verify (e 1 , f 1 ) cannot be invaded by the pure strategy e 2 (and by f 2 in population 2).
We remark that the inequality conditions for a strict NE are independent of population size. Furthermore, the fitness functions (9) when the populations 163 are not monomorphic are convex combinations of the appropriate entries in 164 the time-adjusted payoff bimatrix (e.g., Π e 1 = α π e 11 τ 11 + (1 − α) π e 12 τ 12 for some 165 0 ≤ α ≤ 1). It is the same for the classic bimatrix game except that for 166 us α is no longer a linear function of the strategy frequencies of the other 167 population since the distributional equilibrium is not the standard Hardy-168 Weinberg distribution. In fact, α depends on population size N as well.

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A strict NE can be pictured as corresponding to a particular vertex of 170 the unit square (cf. Figure 2 with the axes scaled to be frequencies of the strict NE, exactly one strict NE (e.g., Figure 2A), or exactly two strict NE 176 that are diagonally opposite each other (e.g., Figure 2E). Furthermore, the 177 classic two-strategy bimatrix game (with nondegenerate payoff bimatrix) can 178 be classified by its strict NE and its interior NE (i.e., its unique NE where 179 both populations are polymorphic) if it exists.

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By examining interior NE, we will see this classification method fails 181 for two-strategy time-constrained bimatrix games (see Section 2.2). These 182 equilibria must satisfy Π e 1 = Π e 2 and Π f 1 = Π f 2 so that neither phenotype 183 can increase its payoff by unilaterally switching its strategy. Unfortunately, 184 obtaining analytic formulas for interior NE seems to be out of reach except 185 in two special cases.

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The other special case is interior symmetric NE (i.e., those on the main 194 diagonal where N e 1 = N f 1 ) for role-independent time constrained bimatrix 195 games. As discussed in Section 2.2, there are up to two such diagonal interior 196 symmetric NE and the formulas for these are given in Křivan and Cressman 197 (2017).

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To find interior NE in the general case, we can instead consider the repli-199 cator equation at fixed population size N. This dynamics is given by 3 where Π e i (N e 1 , N f 1 ) and Π f i (N e 1 , N f 1 ) are fitnesses (9) evaluated at the equi-

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Through the Owner-Intruder game with time-constraints, we illustrate 208 the two special cases mentioned above (i.e., either τ 12 τ 21 = τ 11 τ 22 or interior 209 symmetric NE) as well as the replicator method for the general case.

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3 Replicator dynamics at fixed population size assume that frequencies of e 1 strategists p 1 are described by dp 1 dt (Hofbauer and Sigmund, 1998). Because N e1 = p 1 N and the overall size N of population 1 is assumed to be fixed, we obtain dN e1 dt = dp 1 dt N which yields the first equation in (13).

Owner-Intruder game
The Hawk The following list contains all strict NE of the time-constrained Owner-

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Intruder game (Figure 1). After each item in this list, the panels in Figure 2 233 that have this strict NE are indicated in parentheses.

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In the special case where interaction times satisfy τ 12 τ 21 = τ 11 τ 22 , the interior NE (provided it exists) is given by (12) as .
We observe that when all interaction times are the same, the interior equilib- To investigate interior NE further for the Owner-Intruder game, fitness 256 functions (9) are now 257 Π e 1 = n 11 (V − C) 2τ 11 (n 11 + n 12 ) + n 12 V τ 12 (n 11 + n 12 ) , , , 12 Evaluating these at the equilibrium distribution (6) yields where A is given in (7). To find interior NE, we need to solve Π e 1 = Π e 2 and From extensive simulations of the replicator equation, it seems likely that any interior symmetric NE of two-strategy role-independent time-constrained bimatrix games are always saddles but we have no proof of this conjecture. In the special case where τ 12 τ 21 = τ 11 τ 22 (and τ 12 = τ 21 ), interior symmetric NE are saddles since, from (11), Π e1 (and Π f1 ) depends only on the strategy frequency of the other population, implying that the Jacobian of replicator dynamics (13) evaluated at interior equilibrium (12) has zeros on the main diagonal. This extends the same well-known result for classic role-independent bimatrix games (Hofbauer and Sigmund, 1998     and V > C) and an unstable saddle symmetric interior NE appears. In panel C, Hawk-Hawk interaction time is long enough (τ 11 = 5) that two unstable saddle symmetric interior NE emerge along with two neutrally stable asymmetric ones. Panel D is an asymmetric perturbation of the interaction time matrix from panel C (specifically τ 12 shifts from 1 to 1.1) that perturbs the two asymmetric NE to a stable and unstable one. Since min{τ 12 , τ 21 } > 2τ 22 and V < C in panels F, G, H, (D, D) is the only strict NE. It may be globally asymptotically stable (panel F) or only locally asymptotically stable when there are four interior NE with two unstable saddles and two neutrally stable (the role-independent case of panel G) or two unstable saddles together with one unstable and one stable NE (panel H with perturbed interaction matrix compared to panel G). In the roledependent interaction matrices of panels I and J, τ 12 (respectively τ 21 ) is large enough that the paradoxical ESS (D, H) (respectively (H, D)) is the only strict NE and it is globally asymptotically stable. Finally, panels K and L illustrate that sets of boundary NE emerge (thick black line segments) when V = 1, C = 4, τ 11 = τ 12 = τ 21 = 2τ 22 = 1 (panel K), and V = C (panel L). The number of singles of the two strategies for population 1 are denoted 380 by n e i for i = 1, 2 and for population 2 by n f j for j = 1, 2. Then

Non instantaneous pair formation
are the total number of individuals playing a given strategy. We continue to 382 assume that the total number of individuals in each population is N (i.e., Distributional dynamics of singles and pairs when pair formation is de-385 scribed by the mass action law are then 386 dn e 1 dt = −λn e 1 (n f 1 + n f 2 ) + n 11 τ 11 + n 12 τ 12 dn e 2 dt = −λn e 2 (n f 1 + n f 2 ) + n 21 τ 21 + n 22 τ 22 dn f 1 dt = −λn f 1 (n e 1 + n e 2 ) + n 11 τ 11 + n 21 τ 21 dn f 2 dt = −λn f 2 (n e 1 + n e 2 ) + n 12 τ 12 + n 22 τ 22 dn 11 dt = λn e 1 n f 1 − n 11 τ 11 dn 12 dt = λn e 1 n f 2 − n 12 τ 12 dn 21 dt = λn e 2 n f 1 − n 21 τ 21 dn 22 dt = λn e 2 n f 2 − n 22 τ 22 . ( Appendix C shows that (18) has a unique distributional equilibrium for a 387 fixed N and given N e 1 and N f 1 . 388 11 These last assumptions rule out applying the methods to bimatrix games where newly single individuals may wait after disbanding before they are ready to form new pairs. For example, in the model for parental care of offspring known as the Battle of the Sexes (Dawkins, 1976), when fast females mate with philandering males to produce offspring, it is assumed that the male immediately deserts and begins searching for a new mate whereas the female remains and cares for the offspring for a certain amount of time before searching for a new mate.

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Assuming that singles do not get any payoffs, the fitnesses (i.e., the expected payoff to an individual per unit time) of the four strategies evaluated 390 at the unique equilibrium of (18) are (i, j = 1, 2) These fitness functions depend on N, N e 1 and N f 1 . Since, at the unique 392 distributional equilibrium of (18), fitnesses (19) simplify to (i, j = 1, 2) ) λn e 1 τ 1j + λn e 2 τ 2j + 1 .   The equilibrium distribution of (18) is Substituting these expressions to (21) leads to Thus, up to the positive factor

Evolutionary outcomes with non instantaneous pair formation 419
As we saw in Section 2, evolutionary outcomes of time-constrained bima-420 trix games with instantaneous pair formation depend heavily on pair inter-421 action times when these are not all the same (e.g., Figure 2). This section 422 analyzes the same phenomena when pair formation is not instantaneous. 423 We start by characterizing the strict NE of these games. From (21), at 424 strategy pair (e 1 , f 1 ),

The Owner-Intruder game with non instantaneous pair formation 449
When all interaction times equal to τ as in Section 3.1, there is an interior 450 NE if and only if V < C. As a function of λ and τ , it is given by 23 which is the classic result for the case when V < C.
This is a semi-explicit index 1 differential-algebraic equation (Ascher and 460 Petzold, 1998) that we solve numerically using Mathematica 10.  Figure 2. Panels A and E are identical to their corresponding panels in Figure 2 since these are all equivalent to the classic bimatrix game. There are also no noticeable differences between panels B and F compared to Figure 2. The differences with Figure 2 (which emerges for very large λ) are as follows. For long interaction times between Hawks when V > C, the four interior NE of Figure 2 disappear completely when λ = 1 (panels C' and D') whereas the two asymmetric interior NE become unstable for intermediate λ (panel C). When the interaction time between Doves is short and V < C, the asymmetric interior NE of the role-independent time-constrained bimatrix game lose stability and the two symmetric interior shift apart as λ decreases (panels G and G'). With role-dependent interaction times, the asymptotically stable interior NE of Figure 2H eventually becomes unstable when λ decreases and a stable limit cycle emerges.

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This article extends to two-strategy bimatrix games the new approach In Section 3, this includes the distribution of pairs and singles 13 The difficulty of doing such an analysis can be appreciated by considering the complete analysis for the two-locus two-allele viability selection model of population genetics. Pontz et al. (2018) show that this two-dimensional dynamics on the unit square has at least 192 different phase portraits. We feel our model will have a comparable (or even higher) number of different cases.
play Hawk) even though their payoff by doing so is less than if they split 529 the resource without fighting (i.e., both play Dove) in the classic bimatrix 530 game. 14 The reason is that Hawk strictly dominates Dove in each population.   Writing difference equations for pairs 618 n ij (t + ∆) = n ij (t) − n ij (t) n 11 (t) τ 11 + n 12 (t) τ 12 + n 21 (t) τ 21 + n 22 (t)

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It is not immediately clear that A ≥ 0 where A is given in (7). To see 624 this, expand A as the following quadratic expression in N e 1