Playability properties in games of Deterrence and evolution in the Replicator Dynamics Текст научной статьи по специальности «Математика»

Playability Properties in Games of Deterrence

and

Evolution in the Replicator Dynamics

David Ellison and Michel Rudnianski

LIRSA, CNAM 2 Rue Conté, Paris 75003, France E-mail: michel.rudnianski@cnam.fr

Abstract Since the seminal work of John Maynard Smith (1982), a vast literature has developed on evolution analysis through game theoretic tools. Among the most popular evolutionary systems is the Replicator Dynamics, based in its classical version on the combination between a standard non cooperative matrix game and a dynamic system which evolution depends on the payoffs of the interacting species.

Despite its weaknesses, in particular the fact that it does not take into account emergence and development of species that did not initially exist, the Replicator Dynamics has the advantage of proposing a relatively simple model that analyzes and tests some core features of Darwinian evolution. Nevertheless, the simplicity of the model reaches its limits when one needs to predict accurately the conditions for reaching evolutionary stability. The reason for it is quite obvious: it stems from the possible difficulties to find an analytical solution to the system of equations modelling the Replicator Dynamics.

An alternative approach has been developed, based on matrix games of a different kind, called Games of Deterrence. Matrix Games of Deterrence are qualitative binary games in which selection of strategic pairs results for each player in only two possible outcomes: acceptable (noted 1) and unacceptable (noted 0). It has been shown (Rudnianski, 1991) that each matrix Game of Deterrence can be associated in a one to one relation with a system of equations called the playability system, the solutions of which determine the playability properties of the players’ strategies.

Likewise, it has been shown (Ellison and Rudnianski, 2009) that one could derive evolutionary stability properties of the Replicator Dynamics from the solutions of the playability system associated with a symmetric matrix Game of Deterrence on which the Replicator Dynamics is based.

Thus, it has been established that (Ellison and Rudnianski, 2009):

— To each symmetric solution of the playability system corresponds an evolutionarily stable equilibrium set (ESES)

— If a strategy is not playable in every solution of the playability system, the proportion of the corresponding species in the Replicator Dynamics vanishes with time in every solution of the dynamic system

Keywords: evolutionary games, Games of Deterrence, playability, Replicator Dynamics, species, strategies.

Based on these results, the proposed paper will first extend the analysis already undertaken and propose new results in terms of relations between the solutions of the Game of Deterrence playability system and the solutions of the dynamic system.

The paper will then provide a method for systematically modelling standard matrix games as Games of Deterrence, allowing the previous results to be extended to any standard matrix game. In particular, in certain situations where the standard methods for analyzing dynamic systems do not work, the above bridging between standard games and Games of Deterrence will enable to determine the systems’ asymptotic behaviour.

More precisely, in a first part, after having briefly recalled the definition of the Replicator Dynamics, the paper will recall the definitions and basic properties of Games of Deterrence.

A second part will distinguish between three categories of strategies in the Game of Deterrence under consideration, and will associate specific evolutionary properties with each one.

The third and last part will then develop an algorithm associating a Game of Deterrence with any standard quantitative symmetric matrix game in a way that will enable to generalize the method to the analysis of quantitative evolutionary games.

1. Replicator Dynamics and Games of Deterrence

1.1. Replicator Dynamics

The Replicator Dynamics is a classical dynamic system describing the evolution of a population broken down into several species. The outcome of the interaction between two individuals is given by a symmetric matrix game G.

Moreover, if 0 = (0i,02, ■■■,0n) represents the population’s profile (i.e. 0i is the proportion of species i in the population), then the Replicator Dynamics associated with G is the dynamical system D(G) defined by Qi = 0i(ui — uT) where:

— ui = Y^ 0kuik where uik represents the payoff of species i when interacting with

k

species k

— ut = J2 Qu

i

ui defines the fitness of species i, and it then stems straightforwardly from the above system of differential equations that the evolution of the proportion of a species i in the population depends on the relative fitness of i with respect to the average fitness of the entire population.

The above classical representation of the Replicator Dynamics is equivalent to the following:

— Let Q be the space of population profiles

— Let f be a vector field on Q such that 0 = f (0) with fi(0) = 0i(ui — uT)

An equilibrium of the Replicator Dynamics is then defined as a fixed point of f.

In the following, we will always consider that all species are present in the initial state, i.e. Vi e {1, ■■■, n}, 0i(0) = 0

1.2. Games of Deterrence basic properties

Games of Deterrence consider only two possible states of the world:

— Those which are acceptable for the player under consideration (noted 1)

— Those which are unacceptable for that same player (noted 0)

Given that the players’ objective is to be in an acceptable state of the world, Games of Deterrence analyze the strategies’ playability.

For the sake of simplicity, in the following we shall only consider matrix games, but the definitions that will be introduced extend straightforwardly to N-player games.

Let E and R be two players with respective strategic sets SE (card SE = n) and

Sr (card Sr = p).

We shall consider finite bi-matrix games (SE, Sr, U, V) in normal form where possible outcomes are taken from the set {0,1}. More precisely, for any strategic pair (i, k) e SE x SR, uik and vik define the outcomes for player E and R respectively.

A strategy i of E is said to be safe iff Vk e SR, uik = 1.

A non-safe strategy is said to be dangerous.

Let JE (i) be an index called index of positive playability, such that:

If i is safe then JE (i) = 1

If no^ JE (i) = (1 — jE )(1 — jR ^ n I1 — JR (k)(1 — uik )]

keSn

With jE = El (1 — Je(i)) ; and jr = ]1 (1 — Jr(k))

i^SE kESR

If JE(i) = 1, strategy i e SE is said to be positively playable.

If there are no positively playable strategies in SE, that is if jE = 1, all strategies

i e SE are said to be playable by default.

Similar definitions apply by analogy to strategies k of SR.

A strategy in SE U SR is playable iff it is either positively playable or playable

by default.

The system P of all equations of JE(i),i e SE, JR(k),k e SR, jE and jR is called the playability system of the game.

{0,1} n+p+2 is called the playability set of P

The playability system P may be considered as a dynamic system J = f(J) on the playability set.

A solution of the matrix Game of Deterrence is an element of the playability set which is a solution of P.

It has been shown in (Rudnianski, 1991) that any matrix Game of Deterrence has at least one solution, and that in the general case, there is no uniqueness of the solution.

Given a strategic pair (i, к) є SE x SR, i is said to be a deterrent strategy vis-à-vis к iff the three following conditions apply:

— i is playable

— vik = 0

— Зк' є Sr : Jr (k') = 1

It has been shown (Rudnianski, 1991) that a strategy к є SR is playable iff there is no strategy i є SE deterrent vis-à-vis к. Thus, the study of deterrence properties amounts to analyzing the playability properties of the strategies.

A symmetric Game of Deterrence is a Game of Deterrence (SE,SR,U,V) such that SE = SR and U = V1 (i.e. Vi, к, uik = vki)

In the case of symmetric games, the strategic set will be noted S.

A symmetric solution is a solution in which Vi є S, Je(i) = JR(i)

It has been shown (Ellison and Rudnianski, 2009) that in a symmetric Game of

Deterrence, jE = jR

1.3. Deterrence and evolution

It has been shown (Ellison and Rudnianski, 2009) that for a symmetric Game of

Deterrence G with playability system P and Replicator Dynamics D(G), if:

— P has a symmetric solution for which no strategy is playable by default

— at t = 0, the proportion of each positively playable strategy is greater than the sum of the proportions of the non-playable strategies,

then, whatever the initial profile:

— The proportion of each non-playable strategy decreases exponentially towards zero

— The proportion of each playable strategy has a non-zero limit

This result can be interpreted as follows: each symmetric solution of the playability system is associated with an Evolutionarily Stable Equilibrium Set of the Replicator Dynamics, i.e. the union of the attraction basins of the equilibria is a neighbourhood of the equilibrium set.

2. Further properties of evolutionary Games of Deterrence

2.1. Equivalent strategies and evolution

Definition 1. Two strategies i and j are equivalent if Vk e S,uik = ujk Lemma 1. If i and j are equivalent, then:

— I1 is constant in every solution of the Replicator Dynamics

U3

— i and j have the same playability in every solution of the playability system

Proof. Since strategies i and j are equivalent, u = uj hence (In^Y = (IndiY — (InOj)' = (v,i — ut) — (v,j — ut) = 0

Definition 2. Given a subset X of the strategic set S, let i,k e S,

k is said to be X-dominant vis-à-vis i if Vl e X, u,u < ukl.

Likewise, i and k are said to be X-equivalent if i is X-dominant vis-à-vis k and k

is X-dominant vis-à-vis i.

X-dominance is a reflexive and transitive relation.

2.2. Categorization of playability system solutions

Let G be a symmetric Game of Deterrence with playability system P.

Let & be a function which associates with any given solution a of P a partition

(A, B, C) of the strategic set S of G such that:

— A = {i e S| i is positively playable for both players}

— B = {i e SI i is either positively playable for exactly one player or playable by

default for both players}

— C = {i e SI i is non-playable for both players}

Proposition 1. If a partition (A,B,C) of S verifies:

i e A (uik — 0 k e C) 1 (c

i e C ^ 3k e A : uik =0 J ( )

then (A, B, C) e Im№

Conversely if (A, B, C) e ImiP, then (A, B, C) verifies:

i e A (uik 0 k e C) 1 (c2)

3k e A : uik = 0 ^ i e C !(C 2)

Proof. Let (A, B, C) be a partition of S verifying (C1)

-if A = 0,

Let us consider the following element of the playability set defined by:

— Vi e A, Je(i) = Jn(i) = 1

— Vi e B, Je(i) = 1 and JR(i) = 0

— Vi e C, Je(i) = Jr(i) = 0

— jE = jR = 0

Let us now verify that this element is a solution of P:

It stems from (C1) that:

Vi € A, (1 — jE )(1 — jR ^ n — JR(k)(1 — uik)) = 1 and (1 — jE )(1

kES

Je (k)(1 — uik)) = 1

Vi € C, (1 — jE)(1 — jR) n (1 — JR(k)(1 — uik)) = 0 and (1 — jE)(1 kS

Je (k)(1 — uik)) = 0

It also stems from (C1) that Vi € B, 3k € B : uik = 0.

Indeed, if i € B,i € A and i / C, so 3k € A U C : uik = 0

Hence Vi € B,

(1 — jE)(1 — jR) El (1 — JR(k)(1 — uik)) = 1 and (1 — jE)(1 — jR) FI (1 — JE(k)(1 —

keS keS

uik)) = 0 Also n (1 — JR(k)) = 0 and (1 — Je(k)) = 0

keS keS

The chosen values indeed define a solution a of P, and (A, B, C) = & (a)

-if A = 0,

it stems from the second part of (C1) that C = 0 Hence B = S

Also, it stems from the first part of (C1) that no strategy in S is safe.

Therefore, there is a solution a0 of P in which all strategies are playable by default,

and (A,B,C) = (0,S,0) = &(ao)

Let t be a solution of P and (A, B, C) = &(t),

-if jE = jR = 1 in t, then (A, B, C) = (0, S, 0)

and since no strategy is safe, Vi € S, 3k € S : uik =0 Hence (A,B,C) verifies (C1).

-if jE = jR = 0,

A = {i € S\JE(i) = JR(i) = 1} = {i € S| n (1 — JR(k)(1 — uik)) = n (1 —

keS keS

Je (k)(1 — uik)) = 1}

= {i € S\uik =0 ^ Je(k) = Jr(k) =0} = {i € S\uik =0 ^ k € C}

similarly, C = {i € S\3k € A : uik = 0}

Hence (A,B,C) verifies (C1).

Let a be a solution of P and (A, B, C) = & (a) let i S,

i € A ^ Je(i) = Jr(i) = 1

^ i is safe or (1 — jE)(1 — jR ^ n I1 — JR(k)(1 — uik)] = (1 — jE)(1 — jR ^ n I1 —

keS keS

Je (k)(1 — uik)] = 1

- jR ) El (1 -

keS

- jR) n (1 -

keS

& i is safe or (je = jn = 0 and (uifc =0 ^ Je (k) = Jr (k) = 0))

Yet i is safe ^ (je = jn = 0 and (uifc =0 ^ Je(k) = Jn(k) = 0))

so i e A & (je = jn = 0 and (uifc = 0 ^ Je(k) = Jr(k) = 0))

i e A & (uik =0 ^ k e C)

If 3k e A : uik = 0,

then k is deterrent vis-à-vis i for both players.

Hence i e C

2.3. Categorization of the solutions of the Replicator Dynamics

Let G be a symmetric Game of Deterrence and D(G) its Replicator Dynamics.

Let r be a function which associates with any given solution a of D(G) a partition (A', B', C) of the strategic set S of G such that:

— A' = {i e S\ 6i does not have a zero limit}

— B' = {i e S\ lim 6i = 0 and 9(i) is not integrable}

— C' = {i e S\ 6i is integrable}

Proposition 2. If a solution a of D(G) verifies 1 — uT < <x>,

then (A', B', C') = r(a) verifies:

A' = 0 I

i e A & (uik = 0 ^ k e C ) I (c3)

3k e C' : k is (A' U B')-dominant vis-à-vis i and uik < ukk ^ i e C 'I ( )

3k e C' : k is (A' U B')-dominant vis-à-vis i ^ i e C' J

Proof. Let a be a solution of D(G) such that /J” 1 — uT < <x,

and let (A',B',C') = r(a).

A' = 0 because J2 @i = 1

ÍES

Let i S,

= Ui — ut = (1 — ut) — (1 — Ui) hence 9i(t) = 9i(0)^tt 1-UTe- So 1-Ui

6i(0)e$0 1-UT has a non-zero finite limit,

and e- fo 1-Ui has a finite limit, since it is positive and decreasing so 6i has a limit.

This being true for all i e S, the solution a converges towards an equilibrium.

Also lim 6i = 0 & lim f* 1 — ui =

1 — ui = 1 — ^2 Qkuik = @k(1 — uik) = ^2 @k

kES kES k\uik = 0

hence lim 6i =0 &3k e S : uik = 0 and dk is not integrable

i e A' & (ik e S,uik = 0 ^ k e C')

Let i,k e S such that k is (A' U B')-dominant vis-à-vis i,

let 6c" = Y, Oc,

cEC'

By definition of C', OC' is integrable.

uí — uk = Oi(un — uki) = Oi(uii — uki) + Oi(uii — uki) < Oc'

lES iEC' i/C'

St(t) = ¡i(0)eX--“ < *(<»«*'«• < ê(0)er*c- < +«

is upper-bounded.

Hence, if k e C', then i e C'

Now if k e C' and uik < ukk,

ui — uk < Oc' + (uik — ukk)0k = Oc' — Ok

80 (&y = kM - u*) ^ fcVc' - dk) = etBc, - et

o < J^{t) < 1^(0) + f0 ^6c - f0 0i hence fg 0i < J£(0) + f* 9c

Since is upper-bounded and 6c is integrable, is integrable

hence 0i is integrable, and i e C'

Corollary 1. For any solution a of D(G), let ra : S ^ {A',B' ,C'} be such that

ii e S,i e ra(i) in the partition r(a). Let us equip the set {A',B',C'} with the

alphebetical order: A' > B' > C'.

Let (i,k) eS2. If k is (A' U B')-dominant vis-à-vis i, then ra (k) > ra(i)

Also if i and k are (A' U B')-equivalent, then ra(i) = ra (k)

Proof. Let k be (A' U B')-dominant vis-à-vis i,

If k e C', then it stems from proposition 2 that i e C'

If k e B', then k e A', hence 3l / C' : uki = 0 and since uii < uki, uii = 0 whence i e A'

If k e A', then A' > ra(i)

Hence ra (k) > ra (i)

If i and k are (A' U B')-equivalent, then ra(k) > ra(i) and ra(i) > ra(k)

Hence ra(i) = ra(k)

3. Bridging binary and quantitative games

In a first part, the present section will proceed to a classical analysis of the Replicator Dynamics associated with an elementary example of 2x2 standard game. In a second part, an alternative approach based on the transformation of the standard game into a Game of Deterrence will be developed. The third part will generalize the new approach, which will be applied in the fourth part to a case which the standard approach cannot solve comprehensively.

3.1. Example 1: the standard approach

Let us consider the following symmetric matrix game G in which 0 < a < 1:

G

% k

% (1,1) (1 ,a)

k (a, 1) (0,0)

Let 9 — (9i,9k) G O be the profile of the population.

The average payoffs of the two species are:

Ui = 1 uk — a9i

and uT — 9i + a9i9k

Hence O' — f (9) — (9i(1 — 9i — aOiOk), Ok(aOi — 9i — aOiOk)

It can be seen by the classical analysis of the Replicator Dynamics that in every solution of D(G), 9k decreases exponentially, leading to the equilibrium 9 — (1,0). Indeed, in this simple example, the classical approach enables to completely determine the trajectories, and the equilibria.

3.2. Alternative approach

Let us now introduce the following alternative approach the rationale of which will be justified later.

A possible interpretation of player Column receiving payoff a when the strategic pair (i, k) is selected, is that species i can be divided into two sub-species i1 and i2, such that player Column, when playing species k, gets a payoff of 1 against species

ii, and 0 against species i2, provided that the proportion in species i of ii and i2 is given by (a, 1 — a).

This in turn implies that the dynamics associated with G may be considered equivalent to the dynamics of the following game G' when the ratio of the two sub-species equals

G'

i 1 *2 k

*1 (1,1) (1,1) (1,1)

*2 (1,1) (1,1) (1,0)

k (1,1) (0,1) (0,0)

Let C = (Ci1, Zi2, Ck ) be the profile of the population.

The average payoffs of the three species are:

Vii = 1 Vi2 = 1 vk Cii

and vT = Cii + Ci2 + Cii Ck

Hence the Replicator Dynamics C' = g(C) is such that:

Ci1 = Cil (1 — Ci 1 — Ci2 — Ci 1 Ck )

Ci2 = Ci2 (1 — Ci 1 — Ci2 — Ci 1 Ck )

Ck = Ck (Ci1 — Ci1 — Ci2 — Ci1 Ck)

As it stems from the matrix of G that strategies i1 and i2 are equivalent,

7^ is constant (lemma 1).

Let H be the subset of the set of profiles of D(G') such that (1 — a)Ci1 = aCi2. Since the ratio is constant, H is stable under the dynamics D(G').

Let us then denote by DH(G') the restriction of D(G') to H

let us then define the splitting maps h and h as follows: h : O ^ H

(@i, Sk) ^ (a9i,(1 — a)9i, Ok)

and h : R2 ^ R3

(x, y) ^ (ax, (1 — a)x, y)

It can be easily seen from the above that h ◦ f = g o h

h generates the breakdown of species i into ii and i2 on the set of profiles, while h

does the same on the tangent space of O

This relation translates in terms of flows as follows:

Let and 4'tg be the flows associated with f and g.

h o f(O) = h(O + ft f (O)) = h(O) + ft h o f (O) = h(O) + ft g o h(O) = tg(h(O))

h o = 4g o h

Hence, since h is bijective, D(G) and DH(G') are topologically conjugate.

In other words, the dynamics of G is equivalent to the dynamics of G' restricted to H.

The playability system P' of G' has a unique solution in which strategies i1 and i2 are positively playable while k is not playable for both players. Indeed, strategies i1 and i2 are safe and i2 is deterrent vis-à-vis k.

It then stems from (Ellison and Rudnianski, 2009) that whatever the initial profile: Ci1 and Ci2 have a non-zero limit C2 has a zero limit

Since f and g\H are topologically conjugate, whatever the initial profile 0(0) in G:

6i has a limit equal to 1 has a zero-limit

These conclusions match exactly those drawn from the standard approach.

3.3. Generalization

Let G be a standard symmetric matrix game,

Let M = max uik and m = min un~

Through replacing all the payoffs u^ by their images via the affinity x , we

obtain a game G with payoffs comprised between 0 and 1.

It is well known (Weibull, 1995) that the Replicator Dynamics is invariant under positive affine transformation of payoffs. In this case, it is accelerated by a factor • If / and / denote the vector fields of D(G) and D(G) respectively, the associated flows satisfy the following relation:

i(M-m)t *) = 4f 1

Proposition 3. Given a standard symmetric game G with payoffs comprised between 0 and 1, there is a binary symmetric matrix game G and a subset H of its set of profiles such that the restriction DH(G') of D(G') to H and D(G) are topologically conjugate.

Proof. This demonstration will use an algorithmic construction of the game G.

Let G be a standard symmetric matrix game with strategic set S = {1,..., n}

Let i G S,

let p = card({uki, k G S}U {0, 1}) — 1, let (a0,..., ap) be such that:

0 = a0 < ai < ... < ap = 1 and {ao,..., ap} = {uki, k G S}U {0,1}

Let Gi be the game obtained from G by replacing strategy i with p equivalent

strategies ii,..., ip and by setting the following payoffs:

Vki = Uki, for k,l G S — {i}

vimi = uu, for 1 < m < p,l G S — {i}

vkim = 1 if m < r, where r is such that uki = ar ; and vkim = 0 otherwise, for

k G S — {i}, 1 < m < p

vimim, = 1 if m' < r, where r is such that uii = ar ; and vimim = 0 otherwise, for

1 < m, m' < p

Let Hi be the subset of the set of profiles Qi of Gi defined by the following equations:

p

y1 < m < p, 6im = (am — am-i) J2 °im

m;=i

The strategies ii, ...,ip are equivalent.

Hence, it stems from lemma 1 that Hi is stable under the dynamics D(Gi)

Let hi be the splitting map: hi : O ^ Hi

9 ^ (9l,..., di—^ (ai — ao)9i,..., (ap — ap-l)9i, 9i+l,..., 9n) and hi : Rn ^ Rn+p—i

(x1, ..., xn) ' ^ (xi, ..., xi—i, (ai a0)xi , ..., (ap ap— i)xi, xi+i, ..., xn )

Let 9 G O and k G S — {i},

vk (hi(9)) vk(91, ..., 9 i —1, (ai ao)9i, ..., (ap ap—1)9i, 9i+i, ..., 9n )

p

= ^ 9lvkl + ^ (am am—i)9ivkim

l=i m=i

r

= Y 9lukl + (am — am—i)9i where r is such that ar = uki

l=i m=i

= 9lukl + ar9i

l=i

= uk (9)

hence yk G S — {i}, vk ◦ hi = uk Similarly, for k G {ii,..., ip},vk ◦ hi = u hence, by linearity vT ◦ hi = uT

and if f and fi denote the vector fields of the Replicator Dynamics of G and Gi respectively,

hi o f = fi o hi

Hence the flows are conjugate via hi, i.e. hi o o h

And since hi is a one-to-one correspondance between O and Hi,

D(G) and DHi(Gi) are topologically conjugate via hi.

Also, {vkim, k G S — {i}U {ii,..., ip}, 1 < m < p}c{0,1}

Hence, the splitting of species i reduces by 1 the number of species which, when selected by one player, may generate a non-binary payoff for the other player, unless strategy i already verifies that property, in which case the algorithm does not modify the game.

Let G' = Gi2 be the game obtained from G by successively applying the above

transformation for each strategy of S,

and let H be the corresponding subset of the set of profiles O' of G',

G' is a binary matrix game and D(G) and DH(G') are topologically conjugate.

Consequence: the asymptotic properties of G can be analyzed through G' and its playability system.

As the algorithm is applied to G, each strategy is split into up to n equivalent strategies. Hence, G' may have up to n2 strategies which can be grouped into n

sets of equivalent strategies. Now, it is generally useful to reduce the size of the playabily system. In the case of G', the fact that equivalent strategies have the same playability in every solution (cf. lemma 1) allows us to reduce the playability system. Indeed:

Proposition 4. Let G' be a symmetric Game of Deterrence with strategic set S' = {1,...,i — 1,

ii,..., ip, i + 1,...,n}, where ii,..., ip are equivalent strategies, and let G' be the game obtained from G' by replacing strategies ii, ...,ip by a strategy io and by setting: wkl = vkh ''yk,l = ii-} ..., ip wio k = viikyk = ii-) ..., ip p

Wkio = FI vkim , 'yk = ii-) ...■) ip

m=i

p

Wio io = n vi\im m=i

Let H' = {(Je(1),..., Je(i—1), Je(ii),..., Je(ip), Je(i+1), ...,Je(n),JR(1),..., Jr(i—

^, JR (ii), ..., JR(ip),

JR(i + 1), ...,JR(n),jE ,jR )\Je (ii) = ... = Je (ip) and JR(ii) = ... = Jr (ip)} C {0,1}2n+2p,

let P' and P'' be the playability systems associated with G' and G'' respectively,

H' is stable under P', and the restriction P'H, of P' to H' is topologically conjugate

to P''. H

Proof Let f : {0,1}2n+2p ^ {0,1}2n+2p and f : {0,1}2n+2 ^ {0,1}2n+2 be the

playability systems P' and P'' respectively.

Since, ii,..., ip are equivalent, the components of f corresponding to Je(ii),..., Je(ip) are equal, as are those corresponding to Jr(ii), ...,Jr(ip).

Hence H' is stable under f . (In fact, Imf c H'.)

So P' can be restricted to H'.

Let hi : H' ^ {0,1}2n+2 be such that:

hi : (Je (1),..., Je (i — 1), Je (ii),..., Je (ip), Je (i + 1),..., Je (n), Jr (1),..., JR(i — 1), jR(ii),..., JR(ip), Jr (i + 1),..., JR(n),jE ,3r ) ^ (Je (1),..., Je (i — 1), Je (ii), Je (i + 1),..., Je(n), Jr(1),..., jR(i — 1),

JR(ii), JR(i + ^ ..^ JR(n),jE ,jR)

hi is a bijection.

In order to prove the topological conjugacy, we must verify that hi o f \h, = f o hi

Let (Je (1),..., Je (i — 1), Je (ii),..., Je (ip), Je (i + 1),..., Je (n), Jr (1),..., JR(i — 1), jR(ii),..., JR(ip), Jr(i + 1),..., JR(n),jE,jr) G H', let k = ii,..., ip,

It stems from the construction of G'' that k is safe in G'' iff it is safe in G'

So if k is safe, the components of hi o f \h, and f o hi corresponding to Je(k) and JR(k) are all equal to 1.

Similarly, i0 is safe iff i1,..., ip are all safe.

Let us now suppose that strategy k is dangerous.

The component of f (Je (1),..., Je (i — 1), Je (il), Je (i+1),..., Je (n), Jr (1),..., Jr (i— 1), J^r^iY

Jr(i + 1),..., Jr(n), je,jR) corresponding to Je(k) is:

(1 — jE)(1 — jR.) H (1 — JR(l)(1 — wkl)) X (1 — JR(ii)(1 — Wki0 ))

l=io

= (1 - jE)(1 - jR) H (1 - JR(l)(1 — vkl)) X (1 - JR(i1)(1 - Il vkim Y l=io m=1

= (1 — jE)(1 — jR ) H (1 — JR(l)(1 — vkl)) X H 1 — JR(il)(1 — vkim) l=io m=1

= (1 — jE)(1 — jR^ El (1 — JR(l)(1 — vkl)),

leS'

which is exactly the same component of hiof (Je(1),..., Je(i — 1), Je(i1),..., Je(ip),

Je(i + 1),..., Je(n),

Jr (1),..., Jr (i — 1), Jr('Î1), ..., Jr (ip), Jr (i + 1),..., JR(n),jE ,jR )

Similarly, all other components match.

Hence P’H' and P" are topologically conjugate.

Corollary 2. Let G be a symmetric Game of Deterrence with a strategic set containing several subsets of equivalent strategies.

Let G" be the game obtained by replacing each subset of equivalent strategies by a single strategy as in proposition 4.

Let H' be the subset of the playability set of elements such that any two equivalent strategies have the same playability for both players.

Then, using the notations of proposition 4, Ph' and and P" are topologically conjugate.

Proof. The result stems straightforwardly from the application of proposition 4 to each subset of equivalent strategies.

Remark 1: Since Imf C H', all the solutions of the playability system P' are in H', and restricting P' to H' does not reduce the number of solutions. Thus, solving P'' is equivalent to solving P'.

Remark 2: The above simplification of the playability system also works in the case of non symmetric Games of Deterrence, when either player E or player R has equivalent strategies.

Remark 3: Let G be a symmetric game with payoffs comprised between 0 and 1, and let G'' be the game obtained by first transforming G into G' as in proposition

3, then transforming G' into G' as in proposiiton 4. If G does not have equivalent strategies in its strategic set, then the strategic set of G'' contains the same number of strategies as that of G. Indeed, each strategy is first replaced by a set of equivalent strategies, which is in turn replaced by a single strategy. If there are equivalent strategies in the strategic set of G, we will choose not to regroup those strategies when building G'', so as to maintain the number of strategies.

3.4. Example 2

Let us consider the following example deriving from the one developped in (Ellison and Rudnianski, 2009), in which individuals may adopt one of three possible behaviours:

— A: aggressive

— D: defensive

— N: neutral

Furthermore, let us assume that:

— when two individuals of the same type interact, the outcome for each one is

1, which means that an aggressive individual will not try to attack another aggressive individual (maybe because of the fear of the outcome)

— a defensive type, when encoutering an aggressive individual, will respond by inflicting damages, represented by a payoff 0 < x < 1 for the aggressor, and will get a 0

— when meeting a defensive or a neutral type, the defensive type does not attack, and the outcome pair is (1, 1)

— a neutral type never responds agressively, and receives a payoff 0 < y < 1 when attacked.

G_

A D N

A (1,1) (x,0) (1,2/)

D (0,x) (1,1) (1,1)

N (y A) (1,1) (1,1)

It has been shown (Ellison and Rudnianski, 2009) that in the extreme case where x = y = 0, the profile (1,0, 0), which corresponds to the whole population being aggressive, is an evolutionarily stable equilibrium, and the set of profiles {(0,t, 1 — t), 0 <t < 1}, which are not individually evolutionarily stable, is an evolutionarily stable equilibrium set.

Let us now consider the case where 0 < x, y < 1.

Let G' and G'' be the following matrix games:

G'

A1 A2 D1 D-2 N

At (1,1) (1,1) (1,0) (0,0) (1,1)

A2 (1,1) (1,1) (1,0) (0,0) (1,0)

Dl (0,1) (0,1) (1,1) (1,1) (1,1)

d2 (0,0) (0,0) (1,1) (1,1) (1,1)

N (1,1) (0,1) (1,1) (1,1) (1,1)

G"

Ao Do N

Ao (1,1) (0,0) (1,0)

D0 (0,0) (1,1) (1,1)

N (0,1) (1,1) (1,1)

Let H be the set of profiles in G' such that (1 — x)9Al = xOa2 and (1 — y)@D1 = у$в2 • By proposition 3, D(G) and DH(G') are topologically conjugate.

Let H' be the subset of the playability set of G' comprised of elements such that Ai and A2 on one hand, and Di and D2 on the other hand, have the same playability for both players.

By proposition 4, P’H, and P'' are topologically conjugate.

It can be easily seen from the matrix of G'' that P'' has three solutions:

— (1,0,0,1,0,0,0,0) (Ao is positively playable while D0 and N are not playable for both players)

— (0,1,1,0,1,1,0,0) (D0 and N are positively playable while A0 is not playable for both players)

— (0,0,0,0,0,0,1,1) (all the strategies are playable by default for both players)

Hence, it stems from the topological conjugacy that P' also has three solutions:

— (1,1,0,0,0,1,1,0,0,0,0,0) (Ai and A2 are positively playable while D\, D2 and N are not playable)

— (0,0,1,1,1,0,0,1,1,1,0,0) (Di, D2 and N are positively playable while Ai and A2 are not playable)

— (0,0,0,0,0,0,0,0,0,0,1,1) (all the strategies are playable by default for both players)

The first two of these solutions satisfy the conditions described in section 1.3. Hence D(G') has two evolutionarily stable equilibrium sets:

— ESESi = {(t, 1 — t, 0,0,0), 0 <t < 1} where only species Ai and A2 remain

— ESES2 = {(0,0, tit2, (1 — ti)t2, 1 —t2), 0 <ti,t2 < 1} where only species Di,D2 and N remain

Hence ESESi П H and ESES2 П H are asymptotically stable equilibrium sets in Dh (G')

ESESi ПH = {(x, 1 — x, 0,0,0)} and ASES2 П H = {(0,0, yt2, (1 — y)t2,1 —12), 0 < t2 < 1}

Now DH(G') is topologically equivalent to D(G),

hence (1,0,0) is an evolutionarily stable equilibium and {(0,t, 1 — t), 0 <t< 1} is an evolutionarily stable equilibrium set in D(G).

The results previously established for the game G in the case where x = у = 0 have been extended to all 0 < x, у < 1. The bridging between binary and quantitative games allows us to establish asymptotic properties of evolutionary quantitative games via playability properties of associated Games of Deterrence.

Also, if a solution a of D(G) tends towards the equilibrium (1,0,0), then 0D and 0N decrease exponentially. So Г (a) = ({A}, %, {D,N}).

And if a tends towards {(0,t, 1 — t), 0 <t < 1}, then 6A decreases exponentially.

So r(a) = ({D,N}, 9, {A}).

It can be easily seen from the matrix of G that these two partitions are the only ones which verify condition (C3). In this case, Imr is exactly the set of partitions of S which verify (C3).

3.5. Shortcut

Proposition 5. Let G be a symmetric matrix game.

Let M and m be the maximal and minimal payoffs in G.

Let G be the game obtained by applying the affinity x i—>• to all the payoffs of

G. Let G' be defined as in proposition 3, and G" as in proposition 4.

Then G'' is the game obtained by replacing the maximum payoff by 1 and all other payoffs by 0 in the matrix of G.

Proof. Using the previous notations (uik,vik and wik represent the payoffs in the games G, G' and G'' respectively), we have:

p

wki0 = fi vkim , ^k = i1, ..., ip m=1 p

Wio io = n vilim m=1

and:

vkim = 1 if m < r, where r is such that uki k G S — {i}, 1 < m < p

vimi ' = 1 if m' < r, where r is such that u^

1 < m, m' < p

Hence:

wkio = 1 if uki = 1 and wkio = 0 otherwise wioio = 1 if uii = 1 and wio io = 0 otherwise

As payoff 1 in game G is the image of payoff M in game G, it follows that G'' is obtained by replacing the maximum payoff by 1 and all other payoffs by 0 in the matrix of G.

Proposition 6. Let G be a symmetric matrix game, and let G'' be the game obtained by replacing the maximum payoff by 1 and all other payoffs by 0 in the matrix of G. Let a be a solution of D(G). If:

— the playability system P'' of G'' has a symmetric solution for which no strategy is playable by default

— a is such that at t = 0, the proportion of each strategy of G corresponding to a positively playable^ strategy in G'' is greater than the sum of the proportions of the strategies of G corresponding to non-playable strategies in G'',

then:

— The proportion of each strategy of G corresponding to a non-playable strategy in G'' decreases exponentially towards zero

= ar ; and vkim = 0 otherwise, for = ar ; and vimim = 0 otherwise, for

— The proportion of each strategy of G corresponding to a playable strategy in G'' has a non-zero limit

Proof. Let M and m be the maximal and minimal payoffs in G.

Let G be the game obtained by applying the affinity x to all the payoffs

of G. Let G' and H be defined as in proposition 3, and G'' and H' as in proposition 4.

P'' is topologically conjugate to P''H,, so the symmetric solution of P'' is conjugate to a solution t of P', which is also symmetric.

By applying the result of section 1.3 to t, we obtain that if at t = 0, the proportion of each strategy which is positively playable in t is greater than the sum of the proportions of the non-playable strategies, then the proportion of each positively playable strategy has a non-zero limit, and the proportion of each non-playable strategy decreases exponentially towards zero.

Then, the conclusions about G follow from the topological conjugacy between D(G) and D h(G') and the invariance by affine transformation linking D(G) and D(G).

4. Conclusion

Starting from a symmetric quantitative game G, we have established the following construction:

G —> G —> G' —> G''

such that:

— the payoffs of G are comprised between 0 and 1 and ft- = -m)t

— G' is binary and DH(G') and D(G) are topologically conjugate

— G'' has the same size as G and P''H, and P'' are topologically conjugate

Now, G'' can be constructed directly from G without computing G and G'.

The results obtained in the previous sections thus enable to:

1. overcome the possible difficulties of solving analytically the Replicator Dynamics

2. establish asymptotic properties of solutions of the Replicator Dynamics associated with any standard symmetric matrix game

3. bridge standard quantitative games with Games of Deterrence, thus paving the way for a treatment of optimality issues through acceptability analysis.

References

Ellison, D., Rudnianski, M. (2009). Is Deterrence Evolutionarily Stable. Annals of the International Society of Dynamic games (Bernhard, P., Gaitsgory, V., Pourtaillier, O., eds), Vol. 10, pp. 357-375. Birkhaüser, Berlin.

Hofbauer, J., Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge, UK.

Nowak, M. A. (2006). Evolutionary Dynamics. The Belknap Press of the Harvard University, Cambridge.

Rudnianski, M. (1991). Deterrence Typology and Nuclear Stability: A Game Theoretic Approach. In: Defense Decision Making (Avenhaus, R., Karkar, H., Rudnianski, M., eds), pp 137-168. Springer Verlag, Heidelberg.

Smith, J. M. (1982). Evolution and the Theory of Games. Cambridge University Press. Von Neumann, J., Morgenstern, O. (1947). Theory of Games and Economic Behavior.

Princeton University Press, Boston.

Weibull, J. W. (1995). Evolutionary Game Theory. MIT Press, Boston, MA.