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On the Evolution of Partial Respect for Ownership

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Abstract

An early prediction of game theory was that respect for ownership—“Bourgeois” or \(B\) behavior—can arise as an arbitrary convention to avoid costly disputes. However, its mirror image—the dispute-avoiding “anti-Bourgeois” or \(X\) behavior through which owners concede their property to intruders—also corresponds to an evolutionarily stable strategy (ESS) under the same conditions. It has since been found repeatedly that first finders of valuable resources are frequently left unchallenged in nature, while evidence for ceding property to intruders without a contest is rare at best. An early verbal rationale for the observed rarity of \(X\) was that two individuals employing such behavior over repeated rounds would be interchanging roles repeatedly, a potentially inefficient outcome known as “infinite regress.” This argument was formalized only recently, through a Hawk–Dove model with ownership asymmetry and a fixed probability \(w\) that two individuals meet again. The analysis showed that if \(w\) and the cost of fighting exceed thresholds determined by the costs of assuming and relinquishing ownership, then \(B\) becomes the only stable convention. However, contrary to expectation, and despite the inefficiency of the \(X\) equilibrium, the analysis also showed that “infinite regress” does not invariably render \(X\) unviable. Nevertheless, this model dealt only with ESSs at which respect for ownership is either absolute or entirely absent. Here, we extend the model to allow for polymorphic evolutionarily stable states, and we use it to explore the conditions that favor partial respect for ownership. In this way, we produce an analytic model that predicts a range of degrees of partial respect for ownership, dependent on model parameters. In particular, we identify a pathway through which any degree of respect for ownership can evolve from absolute disrespect under increasing \(w\) with increasing costs of fighting.

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Acknowledgments

This work was partially supported by a grant from the Simons Foundation (#274041 to Mike Mesterton-Gibbons) and a Discovery grant from NSERC to TNS.

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Correspondence to Mike Mesterton-Gibbons.

Appendices

Appendix 1

Here, we present details of the mathematical model described in Sect. 2. Equations (4) are a pair of nonhomogeneous linear difference equations with constant coefficients. A procedure for solving such equations is well known (see, e.g., [10]). Let us define vectors

$$\begin{aligned} x_n(I,J) \!=\! \begin{bmatrix}F_n(I,J)\\ G_n(I,J)\end{bmatrix},\quad q(I,J,b) \!=\! \begin{bmatrix}p_{11}(I,J)a_{ij}^{OO}(b)\\ p_{21}(I,J)a_{ij}^{IO}(b)\end{bmatrix},\quad r(I,J) \!=\! \begin{bmatrix}p_{12}(I,J)a_{ij}^{OI}\\ p_{22}(I,J)a_{ij}^{II}\end{bmatrix} \nonumber \\ \end{aligned}$$
(21)

Then we can rewrite (4) in vector form as

$$\begin{aligned} x_n(I,J) =(1-w)q(I,J,V) + wq(I,J,0) + r(I,J) + wP(I,J)x_{n-1}(I,J). \end{aligned}$$
(22)

The matrix \(wP(I,J)\) has eigenvalues \(\rho _1(I,J) = w\) with eigenvector \((1,1)^T\) and

$$\begin{aligned} \rho _2(I,J) =w\{p_{11}(I,J)-p_{21}(I,J)\} \end{aligned}$$
(23)

with eigenvector \(((p_{11}(I,J)-1)/p_{21}(I,J),1)^T\), where a superscript \(T\) denotes transpose. Because \(w < 1\), it is clear from Table 2 that the absolute value of each eigenvalue of \(wP(I,J)\) must be strictly less than \(1\) in every case. Hence, in the limit as \(n \rightarrow \infty \), the solution of (22) can be shown to converge to

$$\begin{aligned} x(I,J) =\{I-wP(I,J)\}^{-1}\{(1-w)q(I,J,V) + wq(I,J,0) + r(I,J)\} \end{aligned}$$
(24)

where \(I\) is the \(2\times 2\) identity matrix \(\bigl [{\begin{matrix}1&{}0\\ 0&{}1\end{matrix}}\bigr ]\). With

$$\begin{aligned} \{I-wP(I,J)\}^{-1} =\frac{1}{(1-w)(1-\rho _2(I,J))} \begin{bmatrix}1-wp_{22}(I,J)&wp_{12}(I,J)\\ wp_{21}(I,J)&1-wp_{21}(I,J)\end{bmatrix} \end{aligned}$$
(25)

we now obtain \(F(I,J)\) and \(G(I,J)\) in (5) and (6) as the first and second components, respectively, of the vector on the right-hand side of (24).

Appendix 2

For a contest between an initially resident Bourgeois and intruding Hawk, let the random variable \(K_{{\mathrm{nr}}}\) be the number of rounds during which ownership is not respected; let \(K_{{\mathrm{r}}}\) be the number of rounds during which ownership is respected; and let \(K_{{\mathrm{tot}}} = K_{{\mathrm{nr}}}+K_{{\mathrm{r}}}\) be the total number of rounds. The probability that \(K_{{\mathrm{nr}}} \ge k\) is the probability that the \(B\)-strategist wins each inevitable fight and the contest goes to another round \(k-1\) times in succession, or \(\text{ Prob }(K_{{\mathrm{nr}}} \ge k) = (\frac{1}{2}w)^{k-1}\). Thus

$$\begin{aligned} \text{ Prob }(K_{{\mathrm{nr}}} = k) =\text{ Prob }(K_{{\mathrm{nr}}} \ge k) - \text{ Prob }(K_{{\mathrm{nr}}} \ge k+1) =\bigl \{1-\tfrac{1}{2}w\bigr \}(\tfrac{1}{2}w)^{k-1}, \end{aligned}$$
(26)

and so the expected number of rounds during which ownership is not respected is

$$\begin{aligned} E_{{\mathrm{nr}}}=\sum _{k=1}^\infty k\cdot \text{ Prob }(K_{{\mathrm{nr}}} = k) =\bigl \{1-\tfrac{1}{2}w\bigr \}\sum _{k=1}^\infty k\,(\tfrac{1}{2}w)^{k-1} =\frac{2}{2-w}. \end{aligned}$$
(27)

The contest lasts \(k\) rounds with probability \(w^{k-1}(1-w)\). Thus, the expected total number of rounds is

$$\begin{aligned} E_{{\mathrm{tot}}}=\sum _{k=1}^\infty k\cdot \text{ Prob }(K_{{\mathrm{tot}}} = k) =\bigl \{1-w\bigr \}\sum _{k=1}^\infty k\,w^{k-1} =\frac{1}{1-w}, \end{aligned}$$
(28)

so that the expected number of rounds during which ownership is respected is \(1/(1-w)-2/(2-w) = w/((1-w)(2-w))\), which is a proportion

$$\begin{aligned} \theta _r =\frac{w}{2-w} \end{aligned}$$
(29)

of \(E_{{\mathrm{tot}}}\). Let \(\chi _B\) and \(\chi _H = 1-\chi _B\) be the respective proportions of Bourgeois and Hawk in a polymorphic mixture of the two. Then, the probabilities of contests between two \(H\), two \(B\) and an \(H\) and a \(B\) are \(\chi _H^2\), \(\chi _B^2\) and \(2\chi _H\chi _B\), respectively. In a case of two \(H\) or two \(B\), ownership is respected with probability \(0\) or \(1\), respectively. In a case of an \(H\) and a \(B\), it is equally likely that the initial owner is \(H\) and the intruder \(B\) or that the owner is \(B\) and the intruder is \(H\). In the first of these cases, ownership is respected with probability \(1\). In the second case, ownership is respected with probability \(\theta _r\). Hence, the proportion of all interactions in which ownership is respected is

$$\begin{aligned} \rho _i =0 \cdot \chi _H^2 + 1 \cdot \chi _B^2 + \bigl \{\tfrac{1}{2} \cdot 1 + \tfrac{1}{2} \cdot \theta _r\bigr \}\cdot 2\chi _H\chi _B=\chi _B^2 + (1+\theta _r)(1-\chi _B)\chi _B =\frac{\chi _B(2-w \chi _B)}{2-w}. \end{aligned}$$
(30)

As noted in Sect. 3, however, the proportion of Bourgeois in the strategy mix is also the proportion of owner–intruder contests in which the intruder defers to the owner without a fight, that is, \(\chi _B = \rho _c\). Thus (30) and (13) together imply (14).

Appendix 3

Here, we record some expressions that are too unwieldy for Tables 3 and 4 and determine the signs of some eigenvalues. The first component of E7 is

$$\begin{aligned} \xi _1 =\tfrac{(4-w^2)(4\gamma \{2(1-w)-\varepsilon \} + \{3w^2-4\}\gamma ^2 - \varepsilon w\{4(1-w) + 3\varepsilon w - 2(\varepsilon + \gamma )\})}{16\{\gamma (\gamma +\varepsilon w) + (2-w)(1-\varepsilon )w^3\} - 3(\gamma ^2+\varepsilon ^2)w^4 + 4w^2\{(6\gamma ^2-(2-\varepsilon )^2) + w \varepsilon (2\varepsilon + \gamma )\}}. \end{aligned}$$
(31)

The second component of E7 is

$$\begin{aligned} \xi _2 =\tfrac{(2-w)\{(4+3\{\gamma -\varepsilon \})(2-\varepsilon )w^3 - 2(8-3\gamma ^2-(5-3\varepsilon )\gamma - 7\varepsilon + 2\varepsilon ^2)w^2 + 4(2-\varepsilon -\varepsilon ^2-2\gamma \{1-\varepsilon -\gamma \})w-8\gamma (1-\gamma )\}}{16\{\gamma (\gamma +\varepsilon w) + (2-w)(1-\varepsilon )w^3\} - 3(\gamma ^2+\varepsilon ^2)w^4 + 4w^2\{(6\gamma ^2-(2-\varepsilon )^2) + w \varepsilon (2\varepsilon + \gamma )\}}. \end{aligned}$$
(32)

The second eigenvalue for equilibrium E5 is

$$\begin{aligned} r_2 =\tfrac{(4+3\{\gamma -\varepsilon \})(2-\varepsilon )w^3 - 2(8-3\gamma ^2-(5-3\varepsilon )\gamma - 7\varepsilon + 2\varepsilon ^2)w^2 + 4(2-\varepsilon -\varepsilon ^2-2\gamma \{1-\varepsilon -\gamma \})w-8\gamma (1-\gamma )}{4w(1-w)(2-w)(2+w)(2\gamma -\varepsilon )}. \end{aligned}$$
(33)

The eigenvalues for equilibrium E7 are the roots \(r_1\), \(r_2\) of the quadratic equation

$$\begin{aligned} r^2 - a_1r + \xi _1\xi _2a_0 =0, \end{aligned}$$
(34)

where \(\xi _1\) and \(\xi _2\) are defined by (31)–(32),

$$\begin{aligned} a_1 =\tfrac{(2-w)(2+w)(4\gamma \{1-w-\gamma \}^2-2\{2-(3-w)w-2\gamma \}-\{\gamma +\varepsilon \}w\varepsilon ^2)}{2(1-w)(16\{\gamma (\gamma +\varepsilon w) + (2-w)(1-\varepsilon )w^3\} - 3(\gamma ^2+\varepsilon ^2)w^4 + 4w^2\{(6\gamma ^2-(2-\varepsilon )^2) + w \varepsilon (2\varepsilon + \gamma )\})} \end{aligned}$$
(35)

and

$$\begin{aligned} a_0 =\tfrac{(4-3\{\gamma +\varepsilon \})(2-\varepsilon )w^3 - 2(1-\gamma -\varepsilon )(8-3\{\gamma +\varepsilon \})w^2 + 4(2-3\varepsilon -2\gamma \{3-\varepsilon -\gamma \})w+8\gamma (1-\varepsilon -\gamma )}{16(2+w)(1-w)^2(2-w)^2}, \end{aligned}$$
(36)

which has a positive denominator. So when \((\xi _1,\xi _2)\) lies in the interior of \(\varDelta _2\), implying \(\xi _1\xi _2 > 0\), the sign of the eigenvalue product \(r_1r_2 = \xi _1\xi _2a_0\) is determined by the numerator of \(a_0\), which we denote by \(\nu \). It is straightforward to show that \(\partial \nu /\partial \varepsilon = -2w(1-w)(6-5w) - 8 \gamma (1 - w) - 3\gamma w^2(4-w) - 6\varepsilon (2 - w) w^2\), which is clearly negative for \(w < 1\). Therefore, \(\nu \) cannot exceed its value in the limit as \(\varepsilon \rightarrow 0\), which is \(2(1-w-\gamma )(4w\{1-w\}+\gamma \{4(1-w)+3w^2\})\). The second factor of this product is clearly positive. Thus, if \(\gamma > 1-w\), then \(\nu \) has a negative upper bound, implying that \(r_1r_2 < 0\), and hence that \((\xi _1,\xi _2)\) is an unstable saddle point of dynamical system (11).

If, on the other hand, \(\gamma < 1-w\), then \((\xi _1,\xi _2)\) need not be a saddle point—but in that case it always lies outside \(\varDelta _2\). To establish this result, we proceed in two stages. First we note that for \(x_1 = \xi _1\) and \(x_2 = \xi _2\), the expressions inside squiggly brackets in (11a) and (11b) both equal zero, and therefore equal one another. Adding \(W(\xi _1,\xi _2)\) to both sides of the resulting equation and rearranging, we obtain the equation of a line on which \((\xi _1,\xi _2)\) must lie, namely

$$\begin{aligned} c_1x_1 + c_2x_2 + c_3 =0, \end{aligned}$$
(37)

where we define

$$\begin{aligned} \begin{aligned} c_1&=4(1-\varepsilon )w - (4+w^2)\gamma - 2(2-\varepsilon )w^2\\c_2&=4(1-w)(w-\gamma )- (2-3w)w\varepsilon + w^2\gamma \\c_3&=(2-w)(2\{1-w-\varepsilon \} - w\{\gamma -\varepsilon \}). \end{aligned} \end{aligned}$$
(38)

The point \((\xi _1,\xi _2)\) must lie outside \(\varDelta _2\) when the line (37) does not intersect it. The various cases are enumerated in Table 8. Cases 1, 4 and 6 together exclude the dark shaded triangle in Fig. 6a, as indicated by the corresponding labels. Cases 2 and 7 yield no additional useful information, since they correspond to regions in the upper right of the lighter shaded triangle already excluded above; and the other three cases do not correspond to points where \(\gamma > 0\), \(0 < w < 1\).

Table 8 Enumeration of cases in which the line with Eq. (37) fails to intersect \(\varDelta _2\)
Fig. 6
figure 6

Regions of parameter space for which either \((\xi _1,\xi _2)\), which represents a mix of three pure strategies, must lie outside \(\varDelta _2\) (dark shading) or \(a_0\) must be negative (light or no shading). In either case, the labels for different subregions correspond to Table 8, with \(w_c\) and \(w_2\) defined by (3). a Case I: No Dove. Here \((\xi _1,\xi _2)\) is defined by (31)–(32); \(a_0\) is defined by (36); and \(c_1\), \(c_2\), \(c_3\) are defined by (38). b Case II: No anti-Bourgeois. Here \((\xi _1,\xi _2)\) is defined by (44)–(45); \(a_0\) is defined by (51); \(\gamma _1 = 2(1-\varepsilon )^2/(4-3\varepsilon )\); \(w_1 = 2(1-\varepsilon )/(4-3\varepsilon )\); \(w_3 = (1-3\varepsilon /2)/(1-\varepsilon )\); and \(c_1\), \(c_2\), \(c_3\) are defined by (46). Note that (2) implies \(w_1 < w_c < w_3 < w_2\). The labels for different subregions correspond to Table 8

As far as E7 is concerned, it now remains only to show that if \((\gamma ,w)\) belongs to the unshaded quadrilateral in Fig. 6, then either \((\xi _1,\xi _2)\) lies outside \(\varDelta _2\) or \(\nu < 0\). In fact, \(\nu < 0\) throughout this region [even if \((\xi _1,\xi _2)\) lies outside \(\varDelta _2\)]. To see why, we define

$$\begin{aligned} \gamma _L =1-\varepsilon - \{1-\varepsilon /2\}w \end{aligned}$$
(39)

and note that the unshaded region is defined by \(\max (0,\gamma _L \le \gamma \le \min (1-w,1-\varepsilon )\), so that \(\gamma \ge \gamma _L\) throughout it. Furthermore,

$$\begin{aligned} \frac{\partial \nu }{\partial \gamma }\biggr |_{\gamma =\gamma _L}=-2(1-w)(4\{1-\varepsilon \} + w(\varepsilon + 3w\}) - 3\varepsilon w(2\{1-w\} + w^2) \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 \nu }{\partial \gamma ^2} =-4(4\{1-w\} + 3w^2) \end{aligned}$$

are both negative. Therefore, \(\nu \) cannot exceed

$$\begin{aligned} \nu |_{\gamma =\gamma _L}=-2w\varepsilon (2-\varepsilon )(1-w)(2-w), \end{aligned}$$

and hence is negative. The upshot is that if \((\xi _1,\xi _2)\) lies in \(\varDelta _2\), then it must be an unstable saddle point.

Finally, the second eigenvalue for equilibrium E6 is

$$\begin{aligned} r_2 =\tfrac{(4-3\{\gamma +\varepsilon \})(2-\varepsilon )w^3 - 2(1-\gamma -\varepsilon )(8-3\{\gamma +\varepsilon \})w^2 + 4(2-3\varepsilon -2\gamma \{3-\varepsilon -\gamma \})w+8\gamma (1-\varepsilon -\gamma )}{4w(1-w)(2-w)(2+w)(2\gamma +\varepsilon )}. \end{aligned}$$
(40)

The numerator of this expression is identical to \(\nu \), that is, the numerator of \(a_0\) defined by (36); and we have already shown that \(\nu \) is negative whenever \(\gamma \ge \gamma _L\), where \(\gamma _L\) is defined by (39). But E6 lies inside \(\varDelta _2\) only if \(\gamma > \gamma _L\), by Table 3. Moreover, the denominator of \(r_2\) in (40) is clearly positive. Hence \(r_2 < 0\), implying that E6 is an attractor, whenever E6 lies in \(\varDelta _2\).

Appendix 4

Here, we consider a mixture containing proportions \(x_1\), \(x_2\) and \(x_3\) of \(H\), \(B\) and \(D\), respectively, thus excluding \(X\) instead of \(D\). Then, deleting the third row and column from (7), we obtain

$$\begin{aligned} A =\tfrac{1}{2}V \begin{bmatrix} \frac{1- \gamma - w - \varepsilon /2}{1-w}&1-\frac{\gamma +\varepsilon -1}{2-w}&2-\varepsilon \\ \frac{1-w-\gamma }{2-w}&1&\frac{3-w-\varepsilon }{2-w}\\ 0&1-\frac{1}{2-w}&1-\frac{\varepsilon }{2(1-w)} \end{bmatrix} \end{aligned}$$
(41)

in place of (10). In place of (11) we obtain

$$\begin{aligned} \dot{x}_1&=Vx_1\biggl \{\frac{(1-\varepsilon )(1-x_1-x_2)+1}{2} + \frac{(1-\gamma -\varepsilon )x_2}{2(2-w)} -\frac{(\varepsilon +2\gamma )x_1}{4(1-w)} - W(x_1,x_2)\biggr \}\end{aligned}$$
(42a)
$$\begin{aligned} \dot{x}_2&=Vx_2\biggl \{\frac{(1-\varepsilon )(1-x_1-x_2)-(1+\gamma )x_1}{2(2-w)} + \frac{1}{2} - W(x_1,x_2)\biggr \} \end{aligned}$$
(42b)

where

$$\begin{aligned} W(x_1,x_2)=\tfrac{\varepsilon \{(2-w)(2wx_1\{1-x_1-x_2\}-1)+(2-wx_2)x_2\}}{4(1-w)(2-w)}-\gamma x_1\bigl \{\tfrac{x_1}{2(1-w)} +\tfrac{x_2}{2-w}\bigr \} + \tfrac{1}{2}. \end{aligned}$$
(43)

The first component of E7 in Table 5 is

$$\begin{aligned} \xi _1 =\frac{2(1-w)\{2\gamma (1-w)-2(1-\{1-\varepsilon \}w)w + (1+\gamma +\varepsilon )\varepsilon w\}}{4(\gamma ^2-\{1-\varepsilon \}w^2)(1-w)^2 - \varepsilon ^2w^2 + 2\varepsilon \gamma w(1-2w)(2-w)}. \end{aligned}$$
(44)

The second component of E7 in Table 5 is

$$\begin{aligned} \xi _2 =\frac{(2-w)\{4w(1-w)(1-\varepsilon )(1-\gamma -w) - \varepsilon (\varepsilon w + 2\gamma )\}}{4(\gamma ^2-\{1-\varepsilon \}w^2)(1-w)^2 - \varepsilon ^2w^2 + 2\varepsilon \gamma w(1-2w)(2-w)}. \end{aligned}$$
(45)

As in Appendix 3, this equilibrium must lie on a line whose equation has the form (37) with \(c_1 \ldots c_3\) redefined by

$$\begin{aligned} \begin{aligned} c_1&=2w(1-\varepsilon )(1-w) - \varepsilon w - 2\gamma \\ c_2&=2(1-w)((1-\varepsilon )w-\gamma )\\ c_3&=2(1-\varepsilon )(1-w)^2. \end{aligned} \end{aligned}$$
(46)

Because \(c_3\) is invariably positive, the line fails to intersect \(\varDelta _2\) if \(c_1\) and \(c_2\) are both also positive (Case 1 in Table 8); or if \(c_1\) is negative but \(c_2\) and \(c_1 + c_3\) are both positive (Case 4); or if \(c_1\) and \(c_2\) are both negative but \(c_1 + c_3\) and \(c_2 + c_3\) are both positive (Case 6). Collectively, these conditions imply that \((\xi _1,\xi _2)\) lies outside \(\varDelta _2\) whenever \((\gamma ,w)\) lies in the dark shaded triangle in Fig. 6b.

The second eigenvalue for equilibrium E5 in Table 6 is

$$\begin{aligned} r_2 =\dfrac{\varepsilon (1-w)\{2(1-\varepsilon )(1-w) + \varepsilon \} + \{2(\gamma +w-1)+\varepsilon \}\{2w(1-\varepsilon )(1-w) + \varepsilon \}}{8(1-w)(2-w)(\gamma +\varepsilon w)} \end{aligned}$$
(47)

It must be positive—making E5 unstable—whenever the equilibrium lies in \(\varDelta _2\), because \(2(\gamma +w-1)+\varepsilon \) must then be positive, by Table 5.

The second eigenvalue for equilibrium E6 in Table 6 is

$$\begin{aligned} r_2 =\frac{2(\gamma -w^2)(1-\gamma -\varepsilon )+2w\gamma (\gamma +\varepsilon -3) + (2-3\varepsilon )w}{2w(2-w)(\varepsilon + 2\gamma )} \end{aligned}$$
(48)

Let \(\nu \) be its numerator. Then \(\partial \nu /\partial \gamma |_{\gamma =\gamma _L}= -2\{(1-\varepsilon )(1 - w+ w^2) + \varepsilon w\}\) and \(\partial ^2 \nu /\partial \gamma ^2 = -4(1-w)\) are both negative, where \(\gamma _L\) is defined by (39). So \(\nu \) cannot exceed \(\nu |_{\gamma =\gamma _L} = -\frac{1}{2}\varepsilon w(2-\varepsilon )(1-w)(2-w)\), and hence is negative, wherever \(\gamma \ge \gamma _L\). But E6 lies inside \(\varDelta _2\) only if \(\gamma > \gamma _L\), by Table 5. Moreover, the denominator of \(r_2\) is clearly positive. Hence \(r_2 < 0\), implying that E6 is an attractor, if E6 lies in \(\varDelta _2\).

The eigenvalues for equilibrium E7 are the roots \(r_1\), \(r_2\) of the quadratic equation

$$\begin{aligned} r^2 - a_1r + \xi _1\xi _2a_0 =0, \end{aligned}$$
(49)

where \(\xi _1\) and \(\xi _2\) are defined by (44)–(45),

$$\begin{aligned} a_1 =\frac{2\gamma (\varepsilon w + 2(1-w))(1-\gamma -w)-4\gamma \varepsilon (1-w)-w\varepsilon ^2(\varepsilon +3\gamma )}{2\{4(\gamma ^2-(1-\varepsilon )w^2)(1-w)^2 - \varepsilon ^2w^2 + 2\varepsilon \gamma w(1-2w)(2-w)\}} \end{aligned}$$
(50)

and

$$\begin{aligned} a_0 =\frac{2(1-w)(1-\gamma -\varepsilon )(\gamma +w) -w(\varepsilon +2\gamma )}{8(1-w)(2-w)^2}, \end{aligned}$$
(51)

which has a positive denominator. So again as in Appendix 3, when \((\xi _1,\xi _2)\) lies in the interior of \(\varDelta _2\), the sign of the eigenvalue product \(r_1r_2 = \xi _1\xi _2a_0\) is determined by the numerator of \(a_0\), which we denote by \(-\nu \). We have already established that \((\xi _1,\xi _2)\) lies in \(\varDelta _2\) only if \(\gamma \ge \gamma _L\) where \(\gamma _L\) is defined by (39), that is, if \((\gamma ,w)\) lies outside the dark shaded triangle in Fig. 6. Furthermore, \(\partial \nu /\partial \gamma |_{\gamma =\gamma _L} = 2\{(1-\varepsilon )(1-w+w^2)+ \varepsilon w\}\) and \(\partial ^2 \nu /\partial \gamma ^2 = 4(1-w)\) are both positive. Therefore, \(\nu \) must exceed \(\nu |_{\gamma =\gamma _L} = \frac{1}{2}w\varepsilon (2-\varepsilon )(1-w)(2-w)\), and hence is positive. So if \((\xi _1,\xi _2)\) lies in \(\varDelta _2\), then it must be an unstable saddle point.

Appendix 5

Here, we consider a mixture containing three pure strategies, namely \(X\), \(D\) and \(H\) or \(B\) in proportions \(x_1\), \(x_2\) and \(x_3 = 1-x_1-x_2\), respectively. There are two cases, which we will distinguish as Case I or Case II according to whether the third pure strategy is \(H\) or \(B\). Case I and Case II correspond, respectively, to E13 and E14 in Table 7.

Deleting the second row and column from (7) and permuting the remaining rows and columns to make \(H\) the third strategy (not the first, as before), for Case I we obtain

$$\begin{aligned} A =\tfrac{1}{2}V \begin{bmatrix} 1-\frac{\varepsilon }{1-w}&\frac{3+w}{2+w}-\frac{2\varepsilon }{(1-w)(2+w)}&\frac{(1 + w)(1-w-\gamma -\varepsilon )}{(1-w)(2+w)}\\ \frac{1+w}{2+w}\bigl (1-\frac{\varepsilon }{1-w}\bigr )&1-\frac{\varepsilon }{2(1-w)}&0\\ 1-\frac{\gamma +2\varepsilon -1+(\gamma +1)w}{(1-w)(2+w)}&2-\varepsilon&\frac{1- \gamma - w - \varepsilon /2}{1-w} \end{bmatrix} \end{aligned}$$
(52)

in place of (10); whereas in place of (11) we obtain

$$\begin{aligned} \dot{x}_1&=Vx_1\bigl \{\tfrac{(1-w-\varepsilon )(1+w+x_1)-\gamma (1+w)(1-x_1-x_2)+(2-\varepsilon )(1-w)x_2}{2(1-w)(2+w)} - W(x_1,x_2)\bigr \}\end{aligned}$$
(53a)
$$\begin{aligned} \dot{x}_2&=Vx_2\biggl \{\frac{2(1-w-\varepsilon )(1+w)(x_1+x_2)+(2(1-w)+\varepsilon w)x_2}{4(1-w)(2+w)} - W(x_1,x_2)\biggr \} \end{aligned}$$
(53b)

with

$$\begin{aligned} W(x_1,x_2)=\tfrac{\varepsilon ((2+w)(2wx_2(1-x_1-x_2)-1)-(2+wx_1)x_1)}{4(1-w)(2+w)}- \tfrac{\gamma (1-x_1-x_2)(w(1+x_1-x_2)+2(1-x_2))}{2(1-w)(2+w)} + \tfrac{1}{2}. \end{aligned}$$
(54)

The anti-Bourgeois and Dove components of E13 are, respectively,

$$\begin{aligned} \xi _1 =\frac{(2+w)\bigl \{2(2w(1-w)+(1+2w^2)\varepsilon )\gamma - w\bigl (4(1-\varepsilon )(1-w)^2-(1+4w)\varepsilon ^2\bigr )\bigr \} }{4(1+w)^2\gamma ^2 + 2w(2+w)(1+2w)\varepsilon \gamma + w^2q_\varepsilon (w)} \end{aligned}$$
(55)

and

$$\begin{aligned} \xi _2 =\frac{4(1+w)^2\gamma ^2 -4(1-w)(1+w)^2\gamma + 2w(1-w-\varepsilon )(2(1-w-\gamma )+(1+2w)\varepsilon )}{4(1+w)^2\gamma ^2 + 2w(2+w)(1+2w)\varepsilon \gamma + w^2q_\varepsilon (w)} \end{aligned}$$
(56)

where we define

$$\begin{aligned} q_\varepsilon (w) =(4w-1)\varepsilon ^2 - 4(1-\varepsilon )(1-w)^2. \end{aligned}$$
(57)

As in Appendix 3, this equilibrium must lie on a line having Eq. (37) with \(c_1\), \(c_2\), \(c_3\) redefined by

$$\begin{aligned} \begin{aligned} c_1&=2(\gamma (1+w)-w(1-w-\varepsilon )\\ c_2&=2\gamma (1+w)-w(2(1-w)-3\varepsilon )\\ c_3&=2(1+w)(1 - w -\varepsilon - \gamma ) \end{aligned} \end{aligned}$$
(58)

and hence lies outside \(\varDelta _2\) whenever the line fails to intersect it. Cases 1, 4, 5, and 6 of Table 8 together exclude the dark shaded rectangle and triangle in Fig. 7a, as indicated by the corresponding labels; and the other four cases do not correspond to \(\gamma > 0\), \(0 < w < 1\).

Fig. 7
figure 7

Regions of parameter space for which either \((\xi _1,\xi _2)\), which represents a mix of three pure strategies, must lie outside \(\varDelta _2\) (dark or no shading) or \(a_0\) must be negative (light shading). In either case, the labels for different subregions correspond to Table 8, with \(w_c\) and \(w_2\) defined by (3). a Case I: No Bourgeois. Here \((\xi _1,\xi _2)\) is defined by (55)–(56); \(a_0\) is defined by (61); \(w_1\) is defined by (62); \(\overline{w}\) is the smaller root of the quadratic equation \((1-\varepsilon )(1-w)^2 + \varepsilon ^2(1-4w) = 0\), which always lies between \(w_c\) and \(w_2\); and \(c_1\), \(c_2\), \(c_3\) are defined by (58). b Case II: No Hawk. Here \((\xi _1,\xi _2)\) is defined by (70)–(71); \(a_0\) is defined by (75); \(\gamma _1 = \varepsilon (1-\varepsilon )/(1+\varepsilon )\); \(\gamma _2 = \varepsilon (3-2\varepsilon )/(2(1+\varepsilon ))\); \(w_1\) is defined by (76); \(\overline{w}\) is the only root of the cubic equation \(4(1-w)-(2-w)(1+w)^2\varepsilon = 0\), which always lies between \(w_c\) and \(w_2\); \(w_3\) is defined by (68); and \(c_1\), \(c_2\), \(c_3\) are defined by (72). The two dashed curves in the unshaded region are \(\gamma = \gamma _c\) (toward the left) and \(\gamma = \gamma _\pm \) (toward the base), with \(\gamma _c\) defined by (79) and \(\gamma _\pm \) by (80). For the sake of clarity, both diagrams are drawn for \(\varepsilon = \frac{1}{3}\) (which may be unrealistically high)

The eigenvalues for equilibrium E13 are the roots \(r_1\), \(r_2\) of the quadratic equation

$$\begin{aligned} r^2 - a_1r + \xi _1\xi _2a_0 =0, \end{aligned}$$
(59)

where \(\xi _1\) and \(\xi _2\) are defined by (70)–(71),

$$\begin{aligned} a_1 =\frac{(1\!+\!w)\{(4(1\!-\!w)^2 \!+\! (2(3w-2)\!+(\varepsilon \!-\!2w)w)\varepsilon )\gamma \!-\! 2(2(1\!-\!\varepsilon )-(2\!-\varepsilon )w)\gamma ^2 \!-\!\varepsilon ^3 w\}}{2(1\!-\!w)(4(1\!+\!w)^2\gamma ^2 \!+\! 2w(2\!+\!w)(1\!+\!2w)\varepsilon \gamma \!+\! w^2q_\varepsilon (w))}\nonumber \\ \end{aligned}$$
(60)

and

$$\begin{aligned} \begin{aligned} a_0&=-\frac{(1+w)(2(1-\varepsilon )-(2-\varepsilon )w)\gamma + w(1-w + \varepsilon w)(2-2w-3\varepsilon )}{8(1-w)^2(2+w)^2}\\&=-\frac{(2-\varepsilon )(1+w)(w_2-w)\gamma + 2w(1-w + \varepsilon w)(w_1-w)}{8(1-w)^2(2+w)^2} \end{aligned} \end{aligned}$$
(61)

with \(w_1\) redefined by

$$\begin{aligned} w_1 =1-\tfrac{3}{2}\varepsilon \end{aligned}$$
(62)

and \(w_2\) defined by (3). When \((\xi _1,\xi _2)\) lies in the interior of \(\varDelta _2\), the sign of the eigenvalue product \(r_1r_2 = \xi _1\xi _2a_0\) is determined by the numerator of \(-a_0\), which we denote by \(\nu \). By inspection, \(\nu \) must be positive, implying \(a_0 < 0\) and hence that E14 is a saddle point or lies outside \(\varDelta _2\), if \(w < w_1\) (\(< w_2)\) and so \((\gamma ,w)\) lies in the lighter shaded region of Fig. 7 below the (dashed) line where \(w = w_1\); or if \(w > w_1\) and

$$\begin{aligned} \gamma >\frac{2(1-w + \varepsilon w)(w-w_1)w}{(2-\varepsilon )(1+w)(w_2-w)} \end{aligned}$$
(63)

so that \((\gamma ,w)\) lies in the upper part of the lighter shaded region.

It remains to show that if \((\gamma ,w)\) lies in the unshaded region of Fig. 7a, then \((\xi _1,\xi _2)\) lies outside \(\varDelta _2\). On the curve that separates the lighter shaded region from the unshaded one, \(\nu \) (the numerator of \(-a_0\)) changes sign from positive to negative as \(\gamma \) decreases. But (61) implies \(\partial \nu /\partial \gamma > 0\) (because \(w < w_2\)). Hence \(\nu < 0\) throughout the region of interest. From (55)–(56) and (61), a straightforward calculation shows that the Hawk component of E13 is

$$\begin{aligned} \xi _3 =1-\xi _1-\xi _2 =\frac{16(1-w)^2(2+w)^2 \nu }{4(1+w)^2\gamma ^2 + 2w(2+w)(1+2w)\varepsilon \gamma + w^2q_\varepsilon (w)}. \end{aligned}$$
(64)

Because \(q_\varepsilon (w_c) = -\varepsilon ^2 < 0\) and \(q_\varepsilon (w_2) = \varepsilon ^2\{8(1-2\varepsilon ) + 7\varepsilon ^2\}/(2-\varepsilon )^2 > 0\) by (2) and (3), \(q_\varepsilon (w)\) has a zero on the interval \((w_c,w_2)\), which we denote by \(\overline{w}\). For \(\overline{w} < w < w_2\), \(q_\varepsilon (w) > 0\). Thus, the denominator of (64) is positive throughout the part of the unshaded region that lies above the (dashed) line where \(w = \overline{w}\) in Fig. 7a, implying \(\xi _3 < 0\) by (64), and hence that E13 lies outside \(\varDelta _2\). For \(w_c < w < \overline{w}\), on the other hand, the denominator of (64) is negative for values of \(\gamma \) that are sufficiently close to \(0\). The corresponding region in Fig. 7a is only small (because the denominator increases so strongly with \(\gamma \)), but we must show that E13 still lies outside \(\varDelta _2\). First assume that (\(w < \overline{w}\) and) \(w \ge w_c\). Then \(w \ge 1-\varepsilon \), implying \(1+4w \ge 5-4\varepsilon \), and \(1-w \le \varepsilon \), implying \(4(1-\varepsilon )(1-w)^2 \le 4(1-\varepsilon )\varepsilon ^2\); and so \(4(1-\varepsilon )(1-w)^2 - (1+4w)\varepsilon ^2 \le 4(1-\varepsilon )\varepsilon ^2-(5-4\varepsilon )\varepsilon ^2 = -\varepsilon ^2 < 0\), implying that \(\xi _1\) has a positive numerator in (55), and hence is negative. Next, to deal with the lowest part of the unshaded region, assume that \(w < w_c\). In this region, \(1-\varepsilon \le \gamma + w \le 1\), and so in particular \(\gamma \ge 1-w-\varepsilon \). But the numerator of \(\xi _1\) is a strictly increasing function of \(\gamma \). Therefore, its value can never be lower than its value at \(\gamma = 1-w-\varepsilon \), which is readily shown to be \((2-\varepsilon )\varepsilon (2+w)(w_2-w) > 0\); and so if the denominator is negative, then so is \(\xi _1\). We have thus established that E13 lies outside \(\varDelta _2\) throughout the unshaded region.

Likewise, deleting the first row and column from (7) and permuting the remaining rows and columns to make \(B\) the third strategy (not the second, as before), for Case II we obtain

$$\begin{aligned} A =\tfrac{1}{2}V \begin{bmatrix} 1-\frac{\varepsilon }{1-w}&\frac{3+w}{2+w}-\frac{2\varepsilon }{(1-w)(2+w)}&1-\frac{\gamma +\varepsilon }{2(1-w)}\\ \frac{1+w}{2+w}\bigl (1-\frac{\varepsilon }{1-w}\bigr )&1-\frac{\varepsilon }{2(1-w)}&1-\frac{1}{2-w}\\ 1-\frac{\gamma +\varepsilon }{2(1-w)}&\frac{3-w-\varepsilon }{2-w}&1 \end{bmatrix} \end{aligned}$$
(65)

in place of (10); and in place of (11) we obtain

$$\begin{aligned} \dot{x}_1&=Vx_1\biggl \{\frac{(\varepsilon -\gamma )(1-x_1-x_2)-2\varepsilon }{4(1-w)} + \frac{(1-w+\varepsilon w)x_2}{2(1-w)(2+w)} + \frac{1}{2} - W(x_1,x_2)\biggr \}\end{aligned}$$
(66a)
$$\begin{aligned} \dot{x}_2&=Vx_2\biggl \{\frac{(1\!-\!w)(1\!-x_1\!-x_2)}{2(2-w)} \!+\! \frac{(1\!-\!w\!-\!\varepsilon )(1\!+\!w)x_1}{2(1\!-\!w)(2\!+\!w)} \!+\! \frac{(2(1\!-\!w)\!-\!\varepsilon )x_2}{4(1\!-\!w)} \!-\! W(x_1,x_2)\biggr \} \end{aligned}$$
(66b)

where

$$\begin{aligned} W(x_1,x_2)=\tfrac{(4\!-\!w^2)\{2(1\!-\!\gamma x_1(1\!-\!x_2)\!+\!\gamma x_1^2 \!-\! \varepsilon x_1 \!-\! w) \!-\! \varepsilon x_2\} \!+\! \varepsilon w x_2\{(1\!-\!x_2)(2\!+\!w)\!-\!2wx_1\}}{4(1\!-\!w)(2\!-\!w)(2\!+\!w)}. \end{aligned}$$
(67)

It is convenient to redefine

$$\begin{aligned} w_3 =\frac{2(2-\varepsilon )}{4-\varepsilon } \end{aligned}$$
(68)

and thus define

$$\begin{aligned} \begin{aligned} N(\gamma )&=(4-w^2)\gamma ^2 - (4-\varepsilon )(w_3-w)(2\gamma - \varepsilon w)\\ D(\gamma )&=(4-w^2)\gamma \{(4-w^2)\gamma +4\varepsilon w\}-(4-\varepsilon )(w_3-w)\{(4-\varepsilon )(w_3-w) + 2\varepsilon w\}w^2 \end{aligned} \end{aligned}$$
(69)

using notation that suppresses the dependence of these expressions on \(w\) and \(\varepsilon \). Then the anti-Bourgeois and Dove components of E14 are

$$\begin{aligned} \xi _1 =(2+w)\bigl \{(4-w^2)(2(1-w)+\varepsilon w)\gamma - (2-\varepsilon )w(w_2-w)\bigl (4(1-w)+\varepsilon (2+w)\bigr )\bigr \}/D(\gamma ) \end{aligned}$$
(70)

and

$$\begin{aligned} \xi _2 =(4-w^2)N(\gamma )/D(\gamma ), \end{aligned}$$
(71)

respectively, where \(w_2\) is still defined by (3). Note that \(w_1 < w_c < w_2 < w_3\) and that \(D\) is strictly increasing with \(D(0) = -(4-\varepsilon )(w_3-w)\{(4-\varepsilon )(w_3-w) + 2\varepsilon w\}w^2 < 0\), so that \(D(\gamma ) = 0\) has a unique positive root. By contrast, \(N(\gamma ) = 0\) may have two positive roots because \(N(0) = (4-\varepsilon )(w_3-w)\varepsilon w > 0\).

As before, the equilibrium E14 must lie on a line whose equation has the form (37) with \(c_1\), \(c_2\), \(c_3\) redefined by

$$\begin{aligned} \begin{aligned} c_1&=(4-w^2)\gamma -w(4(1-w)-(2-w)\varepsilon )\\c_2&=(4-w^2)\gamma -2w(2(1-w)-(2-w)\varepsilon )\\c_3&=(2+w)(2(1-w)-(2-w)(\varepsilon +\gamma )) \end{aligned} \end{aligned}$$
(72)

and E14 lies outside \(\varDelta _2\) whenever the line fails to intersect it. Cases 1, 4, 5, 6 and 7 of Table 8 together exclude the dark shaded rectangle and curvilinear triangle in Fig. 7b, as indicated by the corresponding labels; and the other three cases do not correspond to \(\gamma > 0\), \(0 < w < 1\).

The eigenvalues for equilibrium E14 are the roots \(r_1\), \(r_2\) of the quadratic equation

$$\begin{aligned} r^2 - a_1r + \xi _1\xi _2a_0 =0, \end{aligned}$$
(73)

where \(\xi _1\) and \(\xi _2\) are defined by (70)–(71),

$$\begin{aligned} a_1 =\tfrac{(2-w)(2+w)(\{4(1-\varepsilon )-(2-\varepsilon )(4-\varepsilon -2w)w)\gamma \}-\varepsilon ^3 w)}{2(1-w)(16\{\gamma (\gamma +\varepsilon w) + (2-w)(1-\varepsilon )w^3\} - 4w^2\{2\gamma ^2+(2-\varepsilon )^2 + w \varepsilon (\gamma -2\varepsilon )\} + (\gamma ^2-3\varepsilon ^2)w^4)} \end{aligned}$$
(74)

and

$$\begin{aligned} \begin{aligned} a_0&=-\frac{(4\!-\!w^2)(2(1\!-\!\varepsilon )-(2\!-\!\varepsilon )w)\gamma \!+\! w(2(1\!-\!w) \!+\! \varepsilon w)(4(1\!-\!w\!-\varepsilon )-(2\!-w)\varepsilon )}{16(2\!-w)(1\!-w)^2(2\!+w)^2}\\&=-\frac{(2-\varepsilon )(2-w)(2+w)(w_2-w)\gamma + (4-\varepsilon )(2(1-w) + \varepsilon w)(w_1-w)w}{16(2-w)(1-w)^2(2+w)^2} \end{aligned} \end{aligned}$$
(75)

where

$$\begin{aligned} w_1 =\frac{2(2-3\varepsilon )}{4-\varepsilon } \end{aligned}$$
(76)

in place of (62). Because \(a_0\) has a positive denominator, when \((\xi _1,\xi _2)\) lies in the interior of \(\varDelta _2\), the sign of the eigenvalue product \(r_1r_2 = \xi _1\xi _2a_0\) is determined by the numerator of \(-a_0\), which again we denote by \(\nu \). By inspection, \(\nu \) must be positive, implying \(a_0 < 0\) and hence that E14 is a saddle point or lies outside \(\varDelta _2\) either if \(w < w_1\) (\(< w_2)\), so that \((\gamma ,w)\) lies in the lighter shaded region of Fig. 7b below the dashed line where \(w = w_1\); or if \(w > w_1\) and \(\gamma > \gamma _R\), where we define

$$\begin{aligned} \gamma _R =\frac{(4-\varepsilon )(2(1-w) + \varepsilon w)(w-w_1)w}{(2-\varepsilon )(2-w)(2+w)(w_2-w)}, \end{aligned}$$
(77)

so that \((\gamma ,w)\) lies in the upper part of the lighter shaded region, above \(w = w_1\) but below \(\gamma = \gamma _R\).

It remains to show that if \((\gamma ,w)\) lies in the unshaded region of Fig. 7b, then \((\xi _1,\xi _2)\) lies outside \(\varDelta _2\). Because \(w < w_2\), (75) implies that \(\partial \nu /\partial \gamma > 0\) (where \(\nu \) is the numerator of \(-a_0\)). But \(\nu \) changes sign \(\gamma \) across the curve \(\gamma = \gamma _R\) that separates the lighter shaded region from the unshaded one. Hence \(\nu < 0\) throughout the unshaded region, and it follows from (70)–(71) and (75) that

$$\begin{aligned} \xi _3 =1-\xi _1-\xi _2 =\frac{16(1-w)^2(2-w)^2(2+w)^2 \nu }{D(\gamma )} \end{aligned}$$
(78)

is negative in the unshaded region if \(D(\gamma ) > 0\) or \(\gamma > \gamma _c\), where

$$\begin{aligned} \gamma _c =\frac{w}{(2-w)(2+w)}\biggl \{\sqrt{8(1-w)^2\{1+(1-\varepsilon )^2\} + w(8-5w)\varepsilon ^2\}}-2\varepsilon \biggl \} \end{aligned}$$
(79)

is the only positive root of \(D(\gamma ) = 0\). The curve on which \(\gamma = \gamma _c\) for \(w_c < w < w_2\) in Fig. 7b is shown dashed. Because \(N(\gamma )\) is quadratic with \(N(0) > 0\), the equation \(N(\gamma ) = 0\) has positive roots

$$\begin{aligned} \gamma _\pm =\frac{(4-\varepsilon )(w_3-w) \pm \sqrt{(4-\varepsilon )(w_3-w)(4(1-w)-(2-w)(1+w)^2\varepsilon )}}{(2-w)(2+w)} \end{aligned}$$
(80)

whenever \(w < \overline{w}\), where \(w_3\) is defined by (68) and \(\overline{w}\) is (re)defined to be the only root of the cubic equation \(4(1-w)-(2-w)(1+w)^2\varepsilon = 0\). The equations \(\gamma = \gamma _\pm \) define two branches of a curve, and only on this curve can \(N(\gamma )\) change sign. However, as illustrated by Fig. 7b, this curve enters the unshaded region only for \(\gamma <\varepsilon (1-\varepsilon )/(1+\varepsilon ) < \gamma < \varepsilon , 1-\varepsilon < w \le \overline{w}\), and all of these points fall outside the region where \(D(\gamma ) < 0\). So, because \(N(0) > 0\), it follows from (71) that \(\xi _2 < 0\) for \(D(\gamma ) < 0\); and we already know that \(\xi _3 < 0\) for \(D(\gamma ) > 0\). We have thus established that if \((\gamma ,w)\) lies in the unshaded region, then \((\xi _1,\xi _2)\) lies outside \(\varDelta _2\).

Appendix 6

Here, we consider a mixture containing all four pure strategies \(H\), \(B\), \(X\) and \(D\) in proportions \(x_1\), \(x_2\), \(x_3\) and \(x_4 = 1-x_1-x_2-x_3\), respectively. As noted toward the end of Sect. 3, the strategy mix \(x = (x_1, x_2, x_3, x_4)\) is now represented by a point \((x_1, x_2, x_3)\) inside the tetrahedron \(\varDelta _3\) where \(0 \le x_1, x_2, x_3 \le x_1 + x_2 + x_3\le 1\), which evolves according to the replicator equations \(\dot{x}_i = x_i\{(Ax)_i - x\cdot A x\}\), \(i = 1, \ldots ,3\) with \(A\) defined by (7). The four new possibilities for a rest point \((\chi _1,\chi _2,\chi _3)\) of this dynamical system not already identified in Tables 3 and 5 are those involving both \(X\) and \(D\) at positive frequency, namely \(XD\), \(HXD\), \(BXD\) and \(HBXD\). We have already shown in Appendix 5 that neither \(HXD\) nor \(BXD\) is an attractor, but it remains to investigate \(XD\) and \(HBXD\). Moreover, although in Appendix 3 we found that \(HB\) is an attractor in the absence of Dove, we need to check the stability of E6 afresh with \(D\) added to the mix.

In the case of \(XD\), we have \(\chi _1 = 0 = \chi _2\) with

$$\begin{aligned} \chi _3 =1 + \frac{2(1-\varepsilon -w)}{\varepsilon w}. \end{aligned}$$
(81)

This point is admissible—that is, in \(\varDelta _3\) and distinct from \(X\) or \(D\)—only for \(0 < \chi _3 < 1\) or \(w_c < w < w_2\), where \(w_c = 1-\varepsilon \) and \(w_2 = (1 - \varepsilon )/(1-\varepsilon /2)\) are defined by (3). The eigenvalues of the corresponding Jacobian matrix—that is, the matrix with \(\partial \dot{x}_i/\partial x_j|_{x_1=\chi _1,x_2=\chi _2,x_3=\chi _3}\) in row \(i\) and column \(j\) for \(i, j = 1, \ldots , 3\), \(\chi _1 = 0 = \chi _2\) and \(\chi _3\) defined by (81)—are

$$\begin{aligned} r_1 =\frac{(2-\varepsilon )(w_c-w)(w_2-w)}{2\varepsilon w(1-w)(2+w)}, \end{aligned}$$
(82)

which is always negative,

$$\begin{aligned} r_2 =\frac{w(1-w_c w)(2(w-w_c)+\varepsilon )-(2-\varepsilon )(1+w)(w_2-w)\gamma }{2\varepsilon w(1-w)(2+w)} \end{aligned}$$
(83)

which is negative when

$$\begin{aligned} \gamma >\frac{w(1-w_c w)(2(w-w_c)+\varepsilon )}{(2-\varepsilon )(1+w)(w_2-w)} \end{aligned}$$
(84)

and

$$\begin{aligned} r_3 =\frac{w(2-(1+w_c) w)(4(w-w_c)+(2-w)\varepsilon )-(2-\varepsilon )(2-w)(2+w)(w_2-w)\gamma }{4\varepsilon w(1-w)(2-w)(2+w)} \end{aligned}$$
(85)

which is negative when (18) holds. The expression on the right-hand side of (18) exceeds the expression on the right-hand side of (84) by

$$\begin{aligned} \frac{w^2(w-w_c)\{2(1+w)(1-w_c w) + 3(2-\varepsilon )(w_2-w)\}}{(2-\varepsilon )(2-w)(1+w)(2+w)(w_2-w)}, \end{aligned}$$

which is positive. Hence, the condition for \(XD\) to be an attractor is that (18) holds (with \(w_c < w < w_2\)).

In the case of \(HB\), from Table 3, we have

$$\begin{aligned} \chi _1 =\dfrac{2(1-w)(1-\gamma -\varepsilon )}{(2\gamma +\varepsilon )w},\quad \chi _2 =1-\dfrac{2(1-w)(1-\gamma -\varepsilon )}{(2\gamma +\varepsilon )w} \end{aligned}$$
(86)

with \(\chi _3 = 0\), and the point \((\chi _1,\chi _2,\chi _3)\) is admissible—that is, in \(\varDelta _3\) and distinct from \(H\) or \(B\)—if both \(\gamma < 1-\varepsilon \) and \((2-\varepsilon )w > 2(1-\gamma -\varepsilon )\). The corresponding Jacobian matrix has \(\partial \dot{x}_i/\partial x_j|_{x_1=\chi _1,x_2=\chi _2,x_3=\chi _3}\) in row \(i\), column \(j\) for \(i, j = 1, \ldots , 3\). Its eigenvalues are the roots of the cubic equation

$$\begin{aligned} (r^2 - a_1r + \chi _1\chi _2a_0)(r-r_3) =0, \end{aligned}$$
(87)

where

$$\begin{aligned} a_0 =\frac{2(1-\varepsilon -\gamma )(w^2-\gamma )-2w(1-(3-\varepsilon )\gamma + \gamma ^2)+3\varepsilon w}{8(1-w)(2-w)^2} \end{aligned}$$
(88)
$$\begin{aligned} a_1 =\frac{2(1-\varepsilon -\gamma )(1-\varepsilon -w^2)-((4-\varepsilon )\gamma +\varepsilon ^2)w + 2w\gamma ^2}{2w(2-w)(2+w)(\varepsilon + 2\gamma )} \end{aligned}$$
(89)

and

$$\begin{aligned} r_3 =\tfrac{(4-3\{\gamma +\varepsilon \})(2-\varepsilon )w^3 - 2(1-\gamma -\varepsilon )(8-3\{\gamma +\varepsilon \})w^2 + 4(2-3\varepsilon -2\gamma \{3-\varepsilon -\gamma \})w+8\gamma (1-\varepsilon -\gamma )}{4w(1-w)(2-w)(2+w)(\varepsilon + 2\gamma )} \end{aligned}$$
(90)

all have positive denominators. The numerator of this last expression is identical to that of (36), and we have already established in Appendix 3 that it must be negative for \(\gamma < 1-\varepsilon \) and \((2-\varepsilon )w > 2(1-\gamma -\varepsilon )\). So the eigenvalue \(r_3\) is likewise negative. Let \(\nu _0\) and \(\nu _1\) denote the numerators of (88) and (89), respectively. Then, the product \(r_1r_2\) of the other two eigenvalues has the sign of \(\nu _0\), whereas their sum \(r_1+r_2\) has the sign of \(\nu _1\). But

$$\begin{aligned} \frac{\partial \nu _0}{\partial \gamma }\biggr |_{\gamma =\gamma _L}=2\{(1-\varepsilon )(1-w + w^2) + \varepsilon w\} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 \nu _0}{\partial \gamma ^2} =4(1-w) \end{aligned}$$

are both positive, where \(\gamma _L\) is defined by (39). Therefore, \(\nu _0\) must exceed

$$\begin{aligned} \nu _0|_{\gamma =\gamma _L}=\tfrac{1}{2}\varepsilon (2-\varepsilon )w(1-w)(2-w), \end{aligned}$$

and hence is positive. Similarly, because

$$\begin{aligned}&\frac{\partial \nu _1}{\partial \gamma }\biggr |_{\gamma =\gamma _L}=-2(1-\varepsilon )(1 + w^2) - 3\varepsilon w,\\&\nu _1|_{\gamma =\gamma _L}=-\tfrac{1}{2}\varepsilon (2-\varepsilon )w(1-w)(2-w) \end{aligned}$$

and

$$\begin{aligned} \nu _1|_{\gamma =1-\varepsilon }=-(2-\varepsilon )w \end{aligned}$$

are all negative while

$$\begin{aligned} \frac{\partial ^2 \nu _1}{\partial \gamma ^2} =4w \end{aligned}$$

does not change sign, \(\nu _1\) is invariably negative when \(\gamma _L \le \gamma \le 1-\varepsilon \) and hence when \((\chi _1,\chi _2,\chi _3)\) is admissible, so that \(r_1\), \(r_2\) have a positive product with a negative sum. Thus, all three eigenvalues are negative. We conclude that \(HB\) remains an attractor, even when Dove is added to the mix.

Finally, we turn our attention to E15, for which the proportions of \(H\), \(B\) and \(X\) are \(\omega _1\), \(\omega _2\) and \(\omega _3\), respectively, where

$$\begin{aligned} \omega _i =\dfrac{n_i(\gamma ,w)}{d} \end{aligned}$$
(91a)

for \(i = 1, \ldots , 3\) with

$$\begin{aligned} n_1(\gamma ,w)= & {} -(4-w^2)\{2(1-\varepsilon )(1-w) + \varepsilon \}\gamma ^2\nonumber \\&+\, 2\{2(8 - 7\varepsilon + \varepsilon ^2)w^2 - (32 - 30\varepsilon + 7\varepsilon ^2) w + 2 (8 - 8\varepsilon + \varepsilon ^2)\}\gamma \nonumber \\&-\, \varepsilon w\{2(4 - 5\varepsilon + \varepsilon ^2)w^2 - (16 - 18\varepsilon + 5\varepsilon ^2) w + 2 (4 - 4\varepsilon + 3\varepsilon ^2)\} \end{aligned}$$
(91b)
$$\begin{aligned} n_2(\gamma ,w)= & {} (2 - w)\bigl \{2\{8-7\varepsilon - 2(3-2\varepsilon )w-2(1-\varepsilon )w^2\}\gamma ^2\nonumber \\&-\, \{4(1 - \varepsilon ) w^3 + 4 (2 - \varepsilon )w^2 - (28 - 24\varepsilon - 3\varepsilon ^2) w + 2 (8 - 8\varepsilon + \varepsilon ^2)\}\gamma \nonumber \\&+\, \varepsilon w\{4 (1 - \varepsilon )(1+ w^2) - 4 (2 - 2\varepsilon + \varepsilon ^2) w + 3\varepsilon ^2\}\bigr \} \end{aligned}$$
(91c)
$$\begin{aligned} n_3(\gamma ,w)= & {} (2 + w)\bigl \{2\{2 (1 - \varepsilon )w^2 - 2 (5 - 3\varepsilon ) w + 8 - \varepsilon \}\gamma ^2\nonumber \\&+\, \{4(1 \!- \varepsilon ) w^3 \!- 4 (2 \!-\! \varepsilon ) (3 \!- \varepsilon ) w^2 \!+ (36 \!- 32 \varepsilon \!+\! 15 \varepsilon ^2) w \!-\! 2 (8 \!-\! 8\varepsilon \!+ \varepsilon ^2)\}\gamma \nonumber \\&+\, \varepsilon w\{4 (1 - \varepsilon ) w^2 - 8 (1 - \varepsilon ) w + 4 - 4 \varepsilon + 3\varepsilon ^2\}\bigr \} \end{aligned}$$
(91d)

and

$$\begin{aligned} d= & {} 2\{4(1-\varepsilon )\gamma w^2(1-w)^2 + \varepsilon w \gamma ^2(20-3w^2)+2\gamma \varepsilon ^2 w^2(5-w)\nonumber \\&+\, 3\gamma ^3(4-w^2)+\varepsilon ^3 w^3\}. \end{aligned}$$
(91e)

Here, we find it convenient to use notation that suppresses both the dependence of \(n_i\) on \(\varepsilon \) and the dependence of \(d\) and \(\omega _i\) on \(\gamma \), \(w\) and \(\varepsilon \). It is also convenient to define or redefine

$$\begin{aligned} \gamma _1 =\frac{8(1-\varepsilon ) +\varepsilon ^2}{2-\varepsilon },\quad \gamma _2 =\frac{8(1-\varepsilon ) +\varepsilon ^2}{8-7\varepsilon },\quad \gamma _3 =\frac{8(1-\varepsilon ) +\varepsilon ^2}{8-\varepsilon },\quad \gamma _4 =\frac{2(1-\varepsilon ) +\varepsilon ^2}{2-\varepsilon } \end{aligned}$$
(92)

and

$$\begin{aligned} w_4 =\frac{4(1-\varepsilon ) +3\varepsilon ^2}{2\{2(1-\varepsilon ) + \varepsilon ^2 + \varepsilon \sqrt{1-\varepsilon +\varepsilon ^2}\}},\quad w_5 =\frac{8-15\varepsilon +7\varepsilon ^2}{(1-\varepsilon )(3-2\varepsilon + \sqrt{25-42\varepsilon +18\varepsilon ^2})}. \end{aligned}$$
(93)

Note that (2) implies \(0 < \gamma _3 < \gamma _4 < \gamma _2 < 1\), \(\gamma _1 > 2\) and \(w_5 > w_4 > w_c\).

The equilibrium E15 is relevant only if \(n_i > 0\) for all \(i\). As indicated by Fig. 8a, the contour \(n_1(\gamma ,w) = 0\) runs from \((0,0)\) through \((\varepsilon , 1-\varepsilon )\) to \((\gamma _1,0)\) with \(n_1 < 0\) outside the contour (shaded region) and \(n_1 > 0\) inside it. As indicated by Fig. 8b, the contour \(n_2(\gamma ,w) = 0\) has one branch running from \((0,0)\) to \((0,w_4)\) and another from \((\gamma _2,0)\) through \((\varepsilon , 1-\varepsilon )\) toward the asymptote \(w = w_5\) (as \(\gamma \rightarrow \infty \)), with \(n_2 < 0\) inside the contour (shaded region) and \(n_2 > 0\) outside it. As indicated by Fig. 8c, the contour \(n_3(\gamma ,w) = 0\) runs from \((0,0)\) to \((\gamma _3,0)\), likewise with \(n_3 < 0\) inside the contour and \(n_3 > 0\) outside it.

Fig. 8
figure 8

Regions of parameter space where a \(n_1\), b \(n_2\), c \(n_3\) and d \(T\) are negative (shaded) or positive (unshaded). Here \(\gamma _1 \ldots \gamma _4\), \(w_4\), \(w_5\) are defined by (92)–(93) and \(w_c = 1-\varepsilon \). It is readily verified that \(n_1(0,0) = n_1(\varepsilon , 1-\varepsilon ) = n_1(\gamma _1,0) = n_2(0,0) = n_2(0,w_4) = n_2(\gamma _2,0) = n_2(\varepsilon , 1-\varepsilon ) = n_3(0, 0) = n_3(\gamma _3,0) = 0\) for all \(\varepsilon \)

A sufficient (though not necessary) condition for an equilibrium to be unstable is that the eigenvalues of the Jacobian matrix have a positive sum. This sum equals the trace of the matrix. Denoting it by \(T(\gamma ,w)\), thus again using notation that suppresses a dependence on \(\varepsilon \), it is straightforward to show that

$$\begin{aligned} 4(1-w)d\,T(\gamma ,w)= & {} 2(4-w^2)\{2(1-\varepsilon )(1-w) + \varepsilon \}\gamma ^3\nonumber \\&+\, \{4(1-\varepsilon )w^4 - 2(2-\varepsilon )^2w^3 - (12-20\varepsilon -\varepsilon ^2)w^2 \nonumber \\&+\, (8-10\varepsilon +3\varepsilon ^2)w - 8(2-2\varepsilon +\varepsilon ^2)\}\gamma ^2\nonumber \\&+\, 2\varepsilon w\{2(4 - 3\varepsilon + \varepsilon ^2)w^2 - (2-\varepsilon )(8-3\varepsilon )w + 2 (4 - 4\varepsilon - \varepsilon ^2)\}\gamma \nonumber \\&+\,\{2(2-\varepsilon )^2w - 4(1-\varepsilon )(1+w^2)-5\varepsilon ^2\}\varepsilon ^2w^2. \end{aligned}$$
(94)

As indicated by Fig. 8d, the contour \(T(\gamma ,w) = 0\) runs from \((\gamma _4,0)\) through \((\varepsilon , 1-\varepsilon )\) to \((\varepsilon ,1)\), with \(T < 0\) to the left of the contour and \(T > 0\) to the right. (This contour may look vertical between these last two points but in fact has a slight curvature, which changes sign; for example, (2) ensures that \((\varepsilon , 1-3\varepsilon /4)\) is on the positive side with \(T(\varepsilon , 1-3\varepsilon /4) = 3(2-3\varepsilon )\varepsilon ^5/8\), whereas \((\varepsilon , 1-\varepsilon /2)\) is on the negative side with \(T(\varepsilon , 1-\varepsilon /2) = -\varepsilon ^6\).) The shaded region in 8(b) always contains the shaded region in 8(c): \(n_2 > 0\) implies \(n_3 > 0\). Hence, the region where \(n_1\), \(n_2\) and \(n_3\) are all positive is the intersection of the two unshaded regions in panels (a) and (b), as indicated (again unshaded) in Fig. 9. Although Figs. 8 and 9 are drawn for a particular value of \(\varepsilon \), namely \(\varepsilon = \frac{1}{4}\), the topology of the regions created by the zero contours is independent of \(\varepsilon \). It is clear from this diagram that \(T > 0\) whenever \((\gamma , w)\) lies in the (unshaded) region where \(n_1\), \(n_2\) and \(n_3\) are all positive, since it lies entirely on the positive side of the zero contour for \(T\). Hence, E15 has at least one eigenvalue with a positive real part, and is therefore unstable, whenever it lies inside \(\varDelta _3\).

Fig. 9
figure 9

Region of parameter space where \(n_1\), \(n_2\) and \(n_3\) are all positive (unshaded). Here \(\gamma _1 \ldots \gamma _4\), \(w_4\) are defined by (92)–(93) and \(w_c = 1-\varepsilon \). The zero contours correspond to those in Fig. 8, but are here shown dashed where they pass through the unshaded region, in which \(n_1\), \(n_2\) and \(n_3\) are not all positive. The solid curve running from \((\gamma _4, 0)\) through \((\varepsilon , 1-\varepsilon )\) to \((\varepsilon , 1)\) is the contour \(T = 0\), on which the trace of the Jacobian changes sign from negative to positive as one moves from left to right. This curve intersects the contours \(n_1 = 0\) and \(n_2 = 0\) at \((\varepsilon ,1-\varepsilon )\), where E15 reduces to monomorphic \(X\) (with all three eigenvalues equal to zero), but otherwise lies entirely inside the unshaded region for every value of \(\varepsilon \)

In effect, we establish that E15 is not an attractor inside \(\varDelta _3\) by showing that, for any value of \(\varepsilon \), the minimum value of \(T(\gamma ,w)\) subject to the constraints \(\gamma \ge 0\), \(0 \le w \le 1\), \(n_1(\gamma ,w) \ge 0\), \(n_2(\gamma ,w) \ge 0\) and \(n_3(\gamma ,w) \ge 0\) is zero, with minimizer \((\varepsilon ,1-\varepsilon )\). This minimization is a standard problem of nonlinear programming (see, e.g., [4]), and for any fixed \(\varepsilon \), the solution is readily verified with mathematical software (e.g., by using the \(Mathematica^{\circledR }\) command NMinimize).

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Mesterton-Gibbons, M., Karabiyik, T. & Sherratt, T.N. On the Evolution of Partial Respect for Ownership. Dyn Games Appl 6, 359–395 (2016). https://doi.org/10.1007/s13235-015-0152-4

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