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
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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
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