Journal of Game Theory 2014, 3(1): 11-18 DOI: 10.5923/j.jgt.20140301.03
Possible Utility Functions for Predator-prey Game Nicola Serra Institute of Radiology, Faculty of Medicine and Surgery of Second, University of Naples, (Naples), Italy
Abstract A mathematical model is proposed to describe the interacting behaviour of predator and prey. This model is
based on the utility functions of the competing individuals. Such functions depend on various parameters that suitably describe animal instincts, considering both physical and environmental conditions. Two possible strategies have been considered for each individual: to race or to be quiet. Our results show that the most significant joint strategies (i.e. both animals run or stay quiet) can be interpreted as solutions of Nash equilibrium of a suitably defined game.
Keywords Behavioural model, Prey – predator model, Utility function, Nash equilibrium, Symmetrical and asymmetrical
game
1. Introduction A large part of problems in biology and environmental sciences is focused on the setting of deterministic or stochastic laws describing the evolution of systems formed by two or more interacting individuals. The result of any interaction usually depends on the behaviour adopted by the involved units. The behaviour of such individuals is often described by means of game theory. The game theory elaborated by Von Neumann & Morgenstern in 1944 is a mathematical theory which analyses the decision-making of individuals during competitions, with a twofold purpose: to explain the interaction behaviours and to suggest the “optimal behaviours”. All the different possible choices of a player define the various strategies to be adopted. The use of the game theory in evolutionistic biology started with the contribution of Maynard Smith and Price, 1973 and completed with the concept of Evolutionarily Stable Strategy, (R. Dawkins, 1979). A new interpretation of strategic behaviour was introduced in order to describe the life-and-death struggle between two interacting individuals, from now on viewed as players. The profit gained by a player as consequence of his strategic choices and that of his competitors can thus be interpreted in terms of adaptation values, reproductive success, and resources control. Further advances in this field are given by Maynard Smith 1974, 1976; G.T. Vickers and C. Cannings, 1987; L. Samuelson 2002; N. Wolf and M. Mangel 2007; A. Sih et al. 2007; R. Cressman et al. 2008; R. Cressman, (2009); Sainmont et al. 2013. * Corresponding author:
[email protected] (Nicola Serra) Published online at http://journal.sapub.org/jgt Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved
The aim of this study is to express the interactions between two species of animals living in the same habitat in terms of competitive games. As usual we denote them as predator and prey and use the following labels: e (eater) and f (food). The initial interacting conditions are: the two individuals are both located in a sufficiently small habitat, so that each of them is aware of the presence of the other, and knows that e is able to capture f. Each player can adopt one out of two suitable strategies: to be quiet (0) or to run (1). Hence, four couples of joint choices are allowed: predator and prey are quiet (0,0); the predator is quiet and the prey is running (0,1); the predator is running and the prey is quiet (1,0); both the predator and prey are running (1,1). Labeling respectively by Se and Sf the sets of strategies of predator and prey, we assume that: Se = Sf = S := {0,1}; S2 := {(0,0), (0,1), (1,0), (1,1)}. (1.1) This paper is organized as follows: in Section 2 the utility functions of both predator and prey are defined and some parameters that depend on the environmental and physical conditions of the two individuals are presented.. In Section 3 the utility functions of predator and prey are analysed, and their dependence are studied on the most relevant parameter, i.e. predator hunger. Moreover, the meaning of other parameters is also explained. In Section 4, the solution of predator-prey game is presented, as a Nash equilibrium. Eventually, some examples are given.
2. Utility Functions For each strategy couples belonging to S2 (cf. (1.1)) the utility functions depend on certain parameters that describe the individuals behaviour. In particular, we consider the following 6 parameters, that deal with the state of the players and the environmental conditions:
12
Nicola Serra: Possible Utility Functions for Predator-prey Game
• h ∈ [ 0,1] ⊂ ; it describes the predator hunger ( h = 0 if the predator is sated, h = 1 if he is fully hungry);
• π ∈ [ 0,1] ⊂ ; it describes the predator liking of the prey (π = 0 if the predator likes the prey a little, π = 1 if he likes the
prey a lot); • η ∈ [ 0,1] ⊂ ; it describes the physical efficiency of the predator versus the prey (η = 0 if the predator is physically worst than the prey, η = 1 if he is physically better than the prey); • τ ∈ [ 0,1] ⊂ ; it describes the environmental conditions of the predator with regard to the prey (τ = 0 if the environmental conditions are better for the prey, τ = 1 if they are better for the predator); • ς e , ς f ∈ [1, +∞[ ⊂ ; they describe respectively the predator aggressiveness and the prey prudence (ςe = 1 if the predator is few aggressive, ςf = 1 if the prey is few prudent). Finally, we introduce two Boolean variables, labelled χ and ϑ, also defined as strategic variables. They describe respectively predator and prey’s behaviour in terms of strategies:
0, if the predator is quiet , χ = 1, if the predator is racing
0, if the prey is quiet . 1, if the prey is racing
ϑ=
(2.1)
The utility functions assign a score to every possible mixture of players’ choices and each player has a different utility function:
ue : ue ( χ , ϑ ) ∈ [ −1, +1] uf
( predator utility function ) : u f ( χ , ϑ ) ∈ [ −1, +1] ( prey utility function )
∀χ,ϑ ∈{0,1}.
It is to notice that the functions ue(χ,ϑ) and uf(χ,ϑ) depend on parameters h, π, η, τ, ςe, ςf. Remark 2.1 The utility of the strategy adopted by each player is influenced by that adopted by the other one. Hence, each player has as many conditional utilities in dependence to the allowed strategies of the other player. We assume the existence of the following conditions for ue and uf, ∀χ,ϑ ∈{0,1}:
∂ue (0, ϑ ) ≤ 0; ∂h
∂ue (0, ϑ ) ≤ 0; ∂π
∂ue (0, ϑ ) ≤ 0; ∂η
∂ue (0, ϑ ) ≤ 0; ∂τ
(2.2a)
∂ue (1, ϑ ) ≥ 0; ∂h
∂ue (1, ϑ ) ≥ 0; ∂π
∂ue (1, ϑ ) ≥ 0; ∂η
∂ue (1, ϑ ) ≥ 0; ∂τ
(2.2b)
∂u f
∂u f
∂u f
∂u f
∂h ∂u f ∂h
( χ , 0) ≤ 0; ( χ ,1) ≥ 0;
∂π ∂u f ∂π
( χ , 0) ≤ 0; ( χ ,1) ≥ 0;
∂η ∂u f ∂η
( χ , 0) ≤ 0; ( χ ,1) ≥ 0;
∂τ ∂u f ∂τ
( χ , 0) ≤ 0
(2.2c)
( χ ,1) ≥ 0;
(2.2d)
Moreover, the following inequalities are held:
ue ( 0,1) ≥ ue ( 0, 0 ) ≥ ue (1, 0 ) ≥ ue (1,1)
(if h = 0);
(2.3a)
ue (1, 0 ) ≥ ue (1,1) ≥ ue ( 0,1) ≥ ue ( 0, 0 )
(if h = 1);
(2.3b)
u f ( 0, 0 ) ≥ u f (1,1) ≥ u f (1, 0 ) ≥ u f ( 0,1)
(if h = 0);
(2.3c)
u f (1,1) ≥ u f ( 0, 0 ) ≥ u f ( 0,1) ≥ u f (1, 0 )
(if h = 1);
(2.3d)
u f (0, 0) ≥ 0; u f (1,1) ≥ 0
(∀ h ∈[0,1]).
(2.3e)
Remark 2.2 From (2.3a,b) it is possible to state that a predator gain can also derive from a wrong strategy adopted by the prey. For example, when the predator is sated (h = 0), inequality ue ( 0,1) ≥ ue ( 0, 0 ) holds because the predator may adopt
the strategy of staying quiet. Therefore, he receives a benefit from the wrong choice of the prey that wastes precious energies racing uselessly.
Journal of Game Theory 2014, 3(1): 11-18
Remark 2.3 Eq. (2.3e) shows that the prey utility in the cases (0,0) and (1,1) is positive, since the prey strategy grants more surviving possibilities in these cases. Remark 2.4 A predator wrong strategy gives a benefit to the prey in terms of utility functions. For instance, if the predator is sated (h = 0), from (2.3c) it is possible to derive that u f (1, 0 ) ≥ u f ( 0,1) . Hence, the prey chooses to stay
The following remarks are held for (2.7): • The right side of (2.7) verifies conditions (2.2a,b) and (2.3a,b); • h is more relevant than the other parameters (the predator hunger engraves on the choices of both the predator and the prey); • ς e is less relevant than h but has the same relevance of
quiet, because it seems that the predator won't be able to chase him. Moreover, when h = 0 the prey feels that the predator is not hungry. When h = 1 (predator is hungry), from (2.3d) it is possible to derive that u f (1,1) ≥ u f ( 0, 0 ) .
π (i.e. a very hungry and very aggressive predator catches the prey which entered in its action-ray, also if the predator dislikes the prey; vice versa a very hungry predator considers particularly the desirability of the prey, even if less aggressive; • π is relevant as ς e , and less relevant than h (i.e. when
In the latter case, the best prey strategy is to run. Contrary, to stay quiet is a very risky strategy, since the pray could be easily caught. Therefore, the strategy couple (1,1) is to prefer to (0,0), in case of h = 1. We notice that the utility functions for prey and predator are not unique. Subsequently we suggest an admissible form for ue and uf, characterized by suitable symmetry properties. First of all we assume that the predator utility ue satisfies the following relation: (2.4) ue ( 0, ϑ ) = −ue (1, ϑ ) , ϑ ∈ {0,1} .
the predator is hungry and less aggressive, he starts to seek a suitable prey in its action-ray); • τ e η are less relevant than h and π (i.e. if the predator is hungry and a desirable prey enters its action-ray, he takes into account the relevant physical and environmental conditions which are crucial to start catching); • ς f is relevant as τ and η (i.e. the predator considers
For any utility function
ue* : ue*
( χ ,ϑ ) ∈ ( −∞, +∞ )
the physical and environmental conditions and the prey attitude, besides an unwary prey is easy to chase).
(2.5)
Now we set:
satisfying Eqs. (2.2a,b) and (2.3a,b), it is always possible to consider the following affine transformation: * u= e ( χ , ϑ ) ue ( χ , ϑ ) −
ue*
(1 − χ ,ϑ ) 2
+ ue*
ςT =
( χ ,ϑ ) ,
∀χ,ϑ∈{0,1}.
The term
(2.6)
It is not hard to verify that the utility function defined in (2.6) satisfies the relation (2.4). Let us now introduce the predator utility function in the case of (χ,ϑ) = (1,0):
1 1 0) ue (1,= η + τ + ς T + 1 ς f
13
to (2.7)
ς eς f + 1 ςf
.
h−2 is a normalization term such that, due ςT h + 1
−1 ≤ ue (1, 0) ≤ 1 .
h
π h − 2 (2.7) . + ς e + ς T h + 1
In the same way, we build the predator utility function in the case (1,1): h
π 1 1 h−2 . ue (1,1) ε η +τ + = + ςe + ς T h + 1 ς f ς T + 1
(2.8)
The variable ε is defined according to (2.3a,b) as follows:
ςf ε = ςe + ς f
h (1− h )
,
ε∈[0,1].
(2.9)
In conclusion, due to (2.4), (2.7) and (2.8), the predator utility function can be condensed as follows: h π 1− χ ϑ 1 − h 1 2 = −1) , ∀χ,ϑ∈{0,1}. ue ( χ , ϑ ) ε η +τ + + ςe + ( ς T h + 1 ς f ς T + 1
(2.10)
14
Nicola Serra: Possible Utility Functions for Predator-prey Game
Moreover, it is possible to define a prey utility function for any strategy couple (χ, ϑ)∈S2, likewise for the predator. According to Eqs. (2.2c,d), (2.4c,d,e) we assume that:
u f ( χ ,1) = −u f ( χ , 0 ) .
(2.11)
In analogy to (2.10), for χ,ϑ∈{0,1}, the prey utility function is:
( −1)1− χ (π + χ −1) 1 1 u f ( χ ,ϑ ) = +ς f (1 − η ) + (1 − τ ) + ς e 2ς T + 1
Figure 1. Predator utility functions and switch points with
π
1− χ 2 ( h − χ )( −1)( ) χ −ϑ . ( −1) − 1− χ ) ( 2ς T ( χ − h ) ) + 1 ( −1) (
=
η
=
τ
= 1 and
(a) Figure 2. Prey utility functions and switch point for
ςe
=
ςf
=1
(b)
ςe
=
ςf
= 1 with (a)
π
=
η
=
τ
= 1; (b)
π
=
η
=
τ
=0
(2.12)
Journal of Game Theory 2014, 3(1): 11-18
3. Switch Points of the Utility Functions Let us now analyze the utility functions with regard to the most relevant parameter h. We introduce the predator and prey switch point and describe the meanings of ς e and
ςf
.
When h ranges in [0,1], the utility functions (2.10), for any (χ, ϑ)∈S2, cross in 4 points that we shall call predator switch points. These are illustrated in Figure 1 for a special case. The value of h for which ue ( 0, 0 ) = ue (1, 0 ) will be
denoted as hA and similarly hB is the value of h such that ue (1,1) = ue ( 0,1) . Hence, ( hA , hB ) will be called predator switch range. The endpoint hA and hB define the threshold between satiated state and hungry state for the predator, such that:
h ≥ hB ,
the predator is hungry
h ≤ hA ,
the predator is satiated
h ∈ ( hA , hB ) ,
• The prey strategies (1,1) and (0,0) are preferred to strategies (1,0) and (0,1), because they optimize the prey energy; * • Denote by h the value of h, for which u f ( 0, 0 ) =
u f (1,1) , or equivalently u f (1, 0 ) = u f ( 0,1) . Hence,
h* is the threshold between prey calm state h < h*
(the prey doesn't perceive risk from the predator * presence), and prey attention state h > h (the prey * perceives the risk of the predator attack). When h = h the prey is in an indecision state. Finally, we discuss the meaning of ς e and ς f in this context. The Figures 3 and 4 show respectively the predator and prey utility for various choices of ς e and ς f in the case (1,1). We notice that
the predator is undecided.
Remark 3.2. If h ∈ ( hA , hB ) and prey is stopped, the
predator catches the prey (he discards solution (0,0), preferring (1,0)). Moreover, if the prey runs, the predator would not waste energies to chase the prey (therefore, he discards solution (1,1) preferring (0,1)). The predator acts as a one that is so hungry to reach the fridge and take all the available food (solution (1,0)), but not so hungry to go out and buy some food (solution (0,1)). In Figure 2, it is shown a list of prey utility functions in suitable cases. Remark 3.3
ςe
and
ςf
characterize a
different kind of predator and prey reacting in the sense of the instinctive evaluation of the game. From Figures 3(b) and 4(b), we assume that the growth rate of the prey utility function in case (1,1) is poorly influenced by ς e . A similar result is held from the predator utility function and values of
ςe
ςf
and
. In conclusion we underline that the
ςf
modify the utility curves and
therefore the switch points. For instance, when
ςe
=
ςf
increases, the switch point decreases (see Figure 4(a)). In the same way, further cases can be analyzed by recalling relations (2.4) and (2.11).
(a) Figure 3. Prey utility functions in the case (1,1), as h ∈ [0,1], (a)
15
(b)
ςe
=
ςf
= 1, 2, 3, 4, 10, (b)
ςe
= 1, 2, 3, 4, 10;
ςf
= 1, for
π=η
=
τ
=1
16
Nicola Serra: Possible Utility Functions for Predator-prey Game
(a)
(b)
Figure 4. Predator utility functions in the case (1,1), as h ∈ [0,1], (a) with
η
=
τ
=
ςf
= 1, 2, 3, 4, 10, (b) with
ςf
= 1, 2, 3, 4;
ςe
= 1 (b), for
π
=
=1
4. The Predator-prey Game Solution
Table 1. Values of
In this section we assume that a finite game begins between a predator and a prey as soon as they perceive the presence of the other individual due to a random interaction. As a simplified model, we describe any interaction between predator and prey as a simultaneous non-cooperative game [1, 12, 13]. Moreover, this is a nonzero-sum game (i.e. in the case (0,0) when the predator is sated, he receives an advantage staying firm, as well as the prey) [5]. Hereafter we show that in our model a Nash equilibrium [11, 13, 14] may arise when h∉ ( hA , hB ) . In this case the Nash equilibrium
suggests the best strategies that the two players should adopt.
Moreover, if h∈ ( hA , hB ) , a Nash equilibrium does not appear in this model. Remark 4.1. The predator–prey game just introduced can be both symmetrical and asymmetrical [6], according to the values of the parameters of the predator and prey utility functions. Indeed, apart from case π = η = τ = 0.5 and ς e =
ςf
ςe
and
ςe =ς f
( ue ( χ , ϑ ) , u f ( χ , ϑ ) )
= 1.The Nash equilibrium is obtained when (χ ,ϑ) = (1,1)
ϑ
x
0
1
1
(0.1471; -0.3615)
(0.0043; 0.3615)
0
(-0.1471; 0.3615)
(-0.0043; -0.3615)
When the predator is sated (h≤ hA), with π = η = τ = 0.5 and ς e = ς f = 1 (symmetric game case), we have hA = 0.3797. Assuming that h = 0.1, the corresponding results are given in Table 2. Table 2. As for Table 1, with h = 0.1, π =η =τ = 0.5, and
( hA , hB ) .
h∉
First of all, we assume that the predator is
hungry (h ≥ hB), with π = η = τ = 0.5 and
ςe =ς f
=1
(symmetric game case). We have that hB = 0.4959. If h = 0.5, the values of the utility functions from the (2.4), (2.10) and (2.12) are given in Table 1, where the couple leading to the Nash equilibrium is underlined.
=
ςf
ϑ
an the the the
ςe
= 1.The Nash equilibrium is obtained when (χ ,ϑ) = (0,0)
by which the game is symmetrical, the interactions
between predator and prey can be considered as asymmetrical game, in which suitable choices of parameters define conditions that favour the predator or prey. Let us now discuss two examples in which predator–prey interaction is a symmetrical game. Example 4.1. We consider two cases in which
for h =π =η =τ = 0.5,
x
0
1
1
(-0.6048; -0.1280)
(-0.6640; 0.1280)
0
(0.6048; 0.7869)
(0.6640; -0.7869)
Example 4.2 We now study a case in which h∈ ( hA , hB ) , with π = η = τ
( hA , hB )
= 0.5, and
ςe
=
ςf
= 1, where
= (0.3797, 0.4959).
From Figure 5(a) for h = 0.45, we easily verify the following inequalities: ue (1, ϑ ) ≥ ue ( 0, ϑ ) (see 2.3b));
and ue ( 0,1) ≥ ue (1,1) (see (2.3a)). Concerning the prey,
inequalities (2.3c) are satisfied (see Figure 5(b)). In this case h* = 0.5 > h (= 0.45). We have:
Journal of Game Theory 2014, 3(1): 11-18
(a)
17
(b)
Figure 5. (a) Predator switch range and (b) prey switch point, with
ς
Table 3. As Table 1, with h = 0.45, π =η =τ = 0.5, and e =
ςf
π = η = τ = 0.5, and ς e
=
ςf
=1
=
1.The Nash equilibrium does not exist
ϑ
x
REFERENCES
0
1
1
(0.0911; -0.3263)
(-0.0519; 0.3263)
[1] I. M. Bomze, (1986) Non-cooperative two-person games in Biology: a classification, Int. J. Game Theory 15, 31-57.
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(-0.0911; 0.3992)
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[2]
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The values of the utility functions reported in Table 3 show that in this case the game has not a Nash equilibrium in pure strategies [14].
5. Concluding Remarks This paper deals a preliminary methodological approach to the study of animal behaviours, with particular regard to the struggle between predator and prey, in order to draw a forecast of the behaviour of interactive individuals. The utility functions proposed in this study are conditioned by more parameters that describe the physical conditions, the instinct of the two animals, the environmental conditions and the strategies that they adopt. The predator utility functions are characterized by an interval ( hA , hB ) , with hA and hB
abscissas of switch points (Section 3). If h∉ ( hA , hB ) , there
exists a Nash equilibrium represented by strategy couples which are similar for predator and prey, i.e. if the predator is hungry (h ≥ hB), the best strategy both predator and prey is to run, indeed if the predator is sated (h ≤ hA) , the best strategy both predator and prey is stay quiet. Indeed, if h∈ ( hA , hB )
the predator-prey game has not a Nash equilibrium in pure strategies, i.e. there is no optimum simultaneous strategy for predator and prey. In conclusion we point out that suitable developments of this research will be oriented to define random utility functions and to introduce further parameters that can influence the interactions between predator and prey.
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Nicola Serra: Possible Utility Functions for Predator-prey Game
Mathematics 54, 286–295. [13] A.M. Ramos, & T. Roubtcek, (2007) Nash Equilibria in Noncooperative Predator–prey Games, Applied. Mathematics. Optimization, 56, 211-241. [14] R. W. Rosenthal, (1973), A class of games possessing pure-strategy Nash equilibria, International Journal of Game Theory 2, 65–67. [15] J. Sainmont, U.H. Thygesen, A.W. Visser. (2013), Diel vertical migration arising in a habitat selection game. Theoretical Ecology 6:2, 241-251. .
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