IMPLEMENTATION OF SUBGAME-PERFECT COOPERATIVE AGREEMENT IN AN EXTENSIVE-FORM GAME Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, XIV, 257-272

Implementation of Subgame-Perfect Cooperative Agreement in an Extensive-Form Game*

Denis Kuzyutin1'2, Yulia Skorodumova1 and Nadezhda Smirnova2

1 St. Petersburg State University, 7/9 Universitetskaya nab., Saint Petersburg 199034, Russia, E-mail: d. kuzyutin@spbu. ru 2 HSE University, St. Petersburg School of Mathematics, Physics and Computer Science Soyuza Pechatnikov ul. 16, St. Petersburg, 190008, Russia E-mail: nvsmirnovaflhse. ru

Abstract A novel approach to sustainable cooperation called subgame-perfect core (S-P Core) was introduced by P. Chander and M. Wooders in 2020 for n-person extensive-form games with terminal payoffs. This solution concept incorporates both subgame perfection and cooperation incentives and implies certain distribution of the total players' payoff at the terminal node of the cooperative history. We use in the paper an extension of the S-P Core to the class of extensive games with payoffs defined at all nodes of the game tree that is based on designing an appropriate payoff distribution procedure P and its implementation when a game unfolds along the cooperative history. The difference is that in accordance with this so-called P-subgame-perfect core the players can redistribute total current payoff at each node in the cooperative path. Moreover, a payoff distribution procedure from the P-S-P Core satisfies a number of good properties such as subgame efficiency, non-negativity and strict balance condition.

In the paper, we examine different properties of the P-S-P Core, introduce several refinements of this cooperative solution and provide examples of its implementation in extensive-form games. Finally, we consider an application of the P-S-P Core to the symmetric discrete-time alternating-move model of fishery management.

Keywords: extensive game, sustainable cooperation, subgame-perfect equilibrium, core, payoff distribution procedure, renewable resource extraction.

1. Introduction

A new solution concept for n-person games in extensive form with terminal payoffs that takes into account both cooperation incentives and subgame perfection notion was introduced in (Chander and Wooders, 2020). This solution concept called subgame-perfect core (S-P Core) is based on two specific properties of an extensive-form game and two main assumptions:

— A set of players in the subgame could be smaller than the set of all players in the original extensive-form game. It is worth noting that a specific approach to the subgame definition that takes into account only the "active" players (i.e. the players which have at least one decision node in the subgame), called "A-subgame concept" was elaborated in (Kuzyutin and Romanenko, 1998; Petrosyan and Kuzyutin, 2000; Kuzyutin et al., 2019a).

* Funding: The reported study was funded by RFBR, project number 21-011-44058. https://doi.org/10.21638/11701/spbu31.2021.19

— "Bargaining power" of some coalition (guaranteed or achievable payoff of this coalition) may vary, i.e., be higher or lower as a game unfolds along the history-generated by a strategy profile.

— If a coalition S deviates from the cooperative scenario in a subgame (all players in S should be active in this subgame) the remaining players do not form any

S

SS separate players. Such assumption was used in (Chander and Tulkens, 1997; Chander, 2007) to introduce so-called 7-characteristic function for cooperative games.

— The payoffs are transferable and, hence, the players can promise to redistribute maximal total payoff of the grand coalition between the players (in the corresponding terminal node) to sustain a cooperative scenario.

A natural extension of the subgame-perfect core concept to more broad class of extensive games (namely, when the player's payoffs are determined at all the nodes) was proposed in (Kuzyutin and Smirnova, 2021). This cooperative solution (called y-S-P Core) implies designing an appropriate payoff distribution procedure (PDP) (see, e.g., Petrosyan and Danilov, 1979; Petrosyan and Kuzyutin, 2000; Yeung and Petrosyan, 2012; Haurie et al., 2012; Kuzyutin et al., 2018) that determines a rule of the current total payoffs redistribution at each node along the cooperative path and satisfies a number of good properties.

In the paper, we are mainly focused on the following issues:

— If we compare two closely related cooperative solutions - subgame-perfect core (Chander and Wooders, 2020) and y-S-P Core (Kuzyutin and Smirnova, 2021) - which of them provides more powerful tool to sustain the cooperative agreement?

— Is there a relationship between strong Nash equilibria (SNE) and y-S-P Core for the class of extensive-form games under consideration?

— If y-S-P Core consists of multiple payoff distribution procedures (in general, this is exactly the case due to Prop. 3), what approaches for the refinement/contraction of the y-S-P Core could be proposed?

We provide answers to these questions below. Besides, we demonstrate the y-S-P Core implementation for sustaining a cooperative scenario in the symmetric version of the discrete-time fishery-management model (Levhari and Mirman, 1980; Haurie et al., 2012; Kuzyutin and Smirnova, 2021).

The remainder of the paper is organized as follows: the class of extensive-form games with perfect information is presented in Section 2. In Section 3, we provide the S-P Core definition due to (Chander and Wooders, 2020) as well as an example of extensive game with terminal payoffs and empty S-P Core. Then, in Section 4, we treat the same game as the game with payoffs defined at each node of the game tree

y

y

yy

implementation in the symmetric two-stage fishery-management model in Section 7 and briefly conclude in Section 8.

2. Extensive-form game with perfect information

We consider a finite multistage game in extensive form following (Kuhn, 1953; Petrosyan and Kuzyutin, 2000; Kuzyutin and Romanenko, 1998; Kuzyutin and Smirnova, 2020). First we need to define the basic notations that will be used in the sequel:

— N = {1,..., n} is the finite set of all players.

— K is the game tree with the root (origin) x0 and the set of all nodes P.

— S(x) is the set of all direct successors of the node x, and S-1(y) is the unique predecessor (parent) of the node y = x0 such that y G S(S-1(y)).

— Pj is the set of all decision nodes of the ¿th player (at these nodes the player i chooses the following node), Pj n Pj = 0, for al 1 i, j G N, i = j.

— Pn+1 = {zj }'jl=1 denotes the set of all terminal nodes (final positions), S(zj) = 0

n+1

Vzj G Pn+1. Note that |J P = P-

i=1

— w = (xo,..., xt-1, xt,..., xT) is the history (or the path) in the game tree, xt-1 = S-1(xt), 1 < t < ^ xt = zj G where "time index" t in xt denotes the ordinal number of this node in the path w.

— hi (x) is the payoff of the ith player at node x G P. We suppose that the payoffs are non-negative, that is, hj(x) > 0 for all i G N, and x G P.

Let (n) denote the class of al 1 finite n-person extensive-form games with perfect information (see, e.g., Kuhn, 1953; Petrosyan and Kuzyutin, 2000; Haurie et al., 2012), while rX0 G (n) denotes a game with origin x0.

Since all the solutions we are interested in throughout the paper are attainable when the players restrict themselves to the class of pure strategies we will focus on this class of strategies. The pure strategy u^-) of the ith player for each node x G Pj specifies the next node uj(x) G S(x) which the piayer i has to select at x. Let U denote the set of all pure strategies of the i-th player, U = nieN Uj- Each strategy-profile u = (u1;..., un) G U generates a path w(u) = (x0,..., xt, xi+1,..., xT) = (xo, x1(u),..., xt(u), xt+1(u),..., xt(u)^, where xt+1 = Uj(xt) G S(xt^f xt G Pj, 0 < t < T — 1, xT G Pn+1, and, hence, a vector of the players' payoffs. Lastly, let

Hj(u) = hj(w(u)) = hj(xT(u))

t=0

iu According to (Kuhn, 1953; Petrosyan and Kuzyutin, 2000; Haurie et al., 2012) each decision node xt G P \ Pn+1 generates a sub game rXt with the subgame tree Kt Pjx% i G N the restriction of Pj on the subgame tree KXt, and

uX% i G N, denote the restriction of the ^^h player's pure strategy u(-) in rX0 on PXt. ^^e (subgame) strategy profile uXt = (uX',..., u^) determines the path wXt(uXt) = (xt,xi+1,...,xT) = (xt,xi+1(uXt),...,xT(uXt)) in the subgame and, correspondly, a vector of the player's payoffs in the subgame rXt:

T

HXt (uXt) = hX' (wXt (uXt)) = ^ hj(xt(uXt)). (1)

t =t

Note that (1) essentially differs from the subgame payoff definition that is accepted for the games with terminal payoffs in extensive form (see, e.g., Kuhn, 1953; Petrosyan and Kuzyutin, 2000).

Definition 1. (Nash, 1950) A strategy profile u = (ui; u2, • • •, un) is a Nash Equilibrium (NE) in rxo e GP(n), if Hi(vi,u-i) < Hj(uj,u_), Vv, e Ui; Vi e N. Let NE(rX0) denote the set of all pure strategy Nash equilibria in rx°.

u

librium (SPE) in rX0 e GP(n), if Vx e P \ Pn+i it Ms that ux e NE(Ex), i.e. the restriction of u on each subgame rx forms a NE in this subgame.

3. S-P Core in an extensive game with terminal payoffs

The novel and promising concept of consistent cooperative behavior called subga-me-perfect core was introduced in (Chander and Wooders, 2020) for extensive form games with perfect information (the authors focus on the special type of extensive-form games when the players payoffs are defined and, hence, could be redistributed only in terminal nodes). For the sake of completeness we provide a subgame-perfect core definition below following (Chander and Wooders, 2020). Namely, the player i G N is called active in the subgame rx if this player has at least one decision node in this subgame, i.e. P, n Kx = 0. Similarly, we shall refer to coalition S C N as active coalition in rx (or simply "active at x") if all the players from S are active in rx.

Given coalition S C N let rx0'S denote the induced game, which differs from rxo

only in that coalition S becomes a new player with hS(x) = ^ h^x), x e Pn+i.

ies

The induced subgame rx's is defined in the same manner given that coalition S is active in rx.

Denote by y(S; x), x e P \ Pn+^ S C N, the highest possible subgame-perfect equilibrium payoff of S in the induced game rx's. In an extensive-form game with terminal payoffs a payoff vector (pi,...,pn) is feasible if P = S h,(z) =

ieN ieN

hN (z) for some terminal node z e P^^ Note that for Miy terminal node z e Pn+i ^^^^e ^^^^^s a unique history w = (xo, • • •, z) leading to this node. Following

ze Pn+i as a

history leading to the feasible payoffs vector (pi; • • • given that P, = hN(z).

ieN

Definition 3. (Chander and Wooders, 2020) A feasible payoff vector (pi; • • • ,pn) belongs to the subgame-perfect core (S-P Core) of a game r e GP (n) in ex-

x

w = (x0, • • •, z) ^^^tog to (pi, • • • ,pn), Pi. = hN(z) md for all coalitions S C N

ie N

x

y(s;x) < ^pi. (2)

ieS

Remark 1. Definition 3 implies (see (Chander and Wooders, 2020) for details) that any feasible payoff vector (pi; • • • ,pn) from the subgame-perfect core is efficient, i.e.

^pi = ma^ hi(z) = hN(z*^ (3)

ieN zePn+1 ieN

Moreover, (2) means that no coalition S which is active in any subgame rx

z*

payoff y(S; x) that is higher than its total payoff in a subgame-perfect core vector

The total payoff (3) could be interpreted as a (maximal) cooperative payoff of the grand coalition N while any history (x0, • • •, z*) is a cooperative history. Then,

( p i , • • • , p n )

payoff among the players) that is consistent in a sense of (2) in any subgame rx along any cooperative history. Hence, if the S-P Core of an extensive-form game r e GP (n) with terminal payoffs is non-empty, the players can ensure the sus-

tainability of the cooperative scenario via redistribution of the total payoff hN (z*) z*

Let us use the following example to demonstrate the S-P Core features.

Example 1. (A 2-player extensive-form game with terminal payoffs: S-P Core).

Let n = 2, Pn+i = {zi, • • •, z8}, Pi = {x0, x3, x4, x5, x6}, P2 = {xi, x2}. Firstly, consider this extensive-form game as a game with terminal payoffs, i.e. the players payoffs hi(z) are defined only in terminal nodes z e Pn+^. This payoffs (hi(z), h2(z))T are written in rectangles below each terminal node, while we assume zero players' payoffs for all decision nodes x0, • • •, x6.

The cooperative history w = (x0,x2,x6,z7 = z*), which is marked in bold in fig.l, corresponds to the maximal total (cooperative) payoff 24. This game possesses a unique subgame-perfect equilibrium (the corresponding choices of the players at all decision nodes are marked as dotted lines) that determines a path w = (x0, xi; x3, zi) and payoff vector (14, 8).

Obviously, pi + p2 = 24 according to (3), and the S-P Core is empty since

Hence, the players can not ensure the sustainability of the cooperative scenario by means of only redistributing the cooperative payoff at terminal node z7 = z*.

However, we will use this example to motivate the way of the S-P Core concept extension to more broad class of extensive-form games. Namely, if the players payoffs are determined at all the nodes, and the players can redistribute the total current payoff at every node in the cooperative path, a natural modification of the S-P Core definition allows to ensure a non-emptiness of the (modified or extended) S-P Core for the game in Ex. 1 as well as to provide the algorithm of its implementation via an appropriate payoff distribution procedure (see, for instance (Petrosyan and Danilov, 1979; Petrosyan and Kuzyutin, 2000; Haurie et al., 2012)).

4. Payoff distribution procedure approach and ^-subgame-perfect core

Hereinafter we consider the general case of an extensive-form game rxo e GP (n) when the players payoffs hi(x) are determined at all nodes x e P. Let w = w(u) = (x0 = x0, • • •, xt, • • •, xT) denote a cooperative (history), i.e.

Y({1}; xo) = 14 < pi, y({2}; x2) = 12 < p2.

T

max

«eu

]T Hi(u) = £ Hi(u) = £ ]T hi(xT) = £ hi(a). (4)

ieN

ieN

ieNT=o

îen

Suppose that there exists a unique cooperative path in rX0 G GP (n). Otherwise, we assume that the players use a specific approach (for instance, the PRB

92460578

12100620

14 7 9 11 6 11 15 16

8 9 8 7 6 12 9 7

Fig. 1. 2-person extensive-form game

algorithm proposed in (Kuzyutin and Smirnova, 2020)) to select a unique cooperative path from all the histories w meeting (4). It is worth noting that PRB algorithm was proved to satisfy time consistency property (see, e.g., Petrosyan, 1977; Petrosyan and Danilov, 1979; Petrosyan and Kuzyutin, 2000; Kuzyutin and Smirnova, 2020).

A vector (pXt,..., p;) such that

T

E p.?t = EE hi(XT) = Ehi (^) (5)

¡en ¡ent=t ie»

determines a possible distribution of the total cooperative (subgame) payoff among the players and could be considered as a cooperative solution for the subgame rxt, xt € w.

Let y = {yi(xT)}, i = 1,..., n; t = 0,..., T; xT € w, denote the Payoff Distribution Procedure (PDP) for the cooperative solution (pi,... ,pn) = (pX0,... ,p;0) (see, e.g., Petrosyan and Danilov, 1979; Petrosyan and Kuzyutin, 2000;

Yeung and Petrosyan, 2012; Haurie et al., 2012; Kuzyutin et al., 2018), i.e., a time schedule of actual payments to the players along the cooperative path. Namely, ^i(xT) denotes the actual current payment that the i-th player receives at node xT € ¿¿instead of hi(xT) if the players employ payoff distribution procedure y.

Let us remind several advantageous properties of a PDP y (see (Petrosyan and Danilov, 1979; Petrosyan and Kuzyutin, 2000; Kuzyutin et al., 2018 Kuzyutin and Smirnova, 2020) for details).

Definition 4. (Kuzyutin and Smirnova, 2020) The PDP y for the payoff vector (pi,... ,pn) meeting (5) satisfies the subgame efficiency condition, if for all Xt G w = (xo,..., Xt)

T

E A(xT) = A(xt,xt+i,..., Xt ) = A (wXt) = pxf4. (6)

T = t

Condition (6) for t = 0 is usually called the efficiency in the whole game (Petrosyan and Danilov, 1979; Petrosyan and Kuzyutin, 2000; Kuzyutin and Smirnova. 2020).

Definition 5. (Petrosyan and Danilov, 1979; Petrosyan and Kuzyutin, 2000; Kuzyutin et al., 2018) The PDP y = {,fij(xT)} satisfies the strict balance condition if for each node XT G w, t = 0,..., T

Eyi(XT) = E hi(XT). (7)

¿ew ¿en

Remark 2. The strict balance constraint (7) for a PDP y follows from the subgame efficiency condition (6).

y

yi(xT) > 0 Vi G N Vt = 0,..., T. (8)

y

coalitions S C N have an incentive to follow a cooperative agreement (pXt,... ,pj;4) in each subgame rxt, Xt G w, along the cooperative history. Let us provide the

y

the sake of completeness.

SCN

y xo

node Xt G w, 1 < t < T — 1), but then decides to deviate from the cooperative mode in the subgame rxt (S should be active in rxt). Then, assuming that the remaining players form singletons (see (Chander, 2007; Chander and Wooders, 2020) for discussion), the highest payoff a coalition S could reach in the whole game rX0 is t-i

equal to J] ys(Xt) + Y(S;Xt).

T=o

If one suppose that

t-i t-i T

E ys(Xt) + Y(S; Xt) < E ys(Xt) + E ys(Xt), (9)

T=o T=o T=t

S Xt

t=0

S Xt

T

Y (S; Xt) < E ys (Xt ), Xt G w, t = 0,...,T — 1. (10)

T=t

Note that inequality (10) implies the same sustainability of the cooperative agreement property as inequality (2) in the S-P Core definition.

Definition 6. (Kuzyutin and Smirnova, 2021). The set of all payoff distribution procedures ft meeting (6), (5), (7), (8), and (10) is called the ft-subgame-perfect core (ft-S-P Core) for an extensive-form game rX0 G (n).

Let us use again 2-person extensive-form game from Example 1 to demonstrate a proposed extension of the S-P Core based on the payoff distribution procedure approach.

Example 2. (A 2-player extensive-form game with payoffs defined at each node: ft-S-P Core.)

Hereinafter suppose that the players' payoffs in the extensive-form game rX0 from Ex. 1 are determined at all nodes x G P (this payoffs (hi(x), h2(x))T are written in the game tree while the total payoffs collected along each history are written below corresponding terminal nodes). Constraints (6), (5) for t = 0 take the form

fti(xo) + fti(x2) + fti(xe) + fti(zr) = fti(w) = pi,

ft2(xo) + ft2(x2) + ft2(x6) + ft2(zr) = ft2(W) = P2, Pi + P2 = 24.

Let us write conditions (10), for instance, in the subgame rX2 along the cooperative history for all coalitions S which are active at x2:

Y(N; x2) = 18 < fti(x2) + ft2(x2) + fti(x6) + ft2(x6) + fti(zr) + ft2(zr), Y({1}; x2) = 10 < fti(x2) + fti(x6) + fti(zr), (11)

Y({2}; x2) =7 < ft2(x2) + ft2(x6) + ft2(z7).

Note that the first constraint in (11) is binding due to strict balance conditions

(7).

Straightforward verification shows that the system (5)-(8) and (10) for this

ft

ftft

fii = 15 + £, /32 =9 - £, -1 < £ < 1.

ft

ftft from cooperation: 3i = 15, 32 = 9;

w xo x2 x6 Z7

fti(xt) 4 1 2 8

ft2(xt) 2 4 2 1

Another FJ)P ft' implies that the second player receives all the benefit from cooperation: fti = 14, ft2 = 10;

w xo x2 x6 Z7

fti (xt) 3 1 2 8

ft2 (xt) 3 4 2 1

5. A strategic support of the ,3-S-P Core

If the players redistribute current payoffs at each node along the cooperative path

y

game rX° which differs from the original game rX0 only in that the payoffs hi(xt), i G N, at every node Xt G w in the cooperative path are replaced by yi(xt). This approach was used earlier, in particular, in (Petrosyan and Kuzyutin, 2000) to introduce a "regularized game" for differential and multistage cooperative game and in (Chander and Wooders, 2020) to define a "strategic transform" of a game in an extensive form with terminal payoffs.

y

then there exists a SPE u in a non-cooperative related game that generates exactly a cooperative path w = w(u) with a SPE payoffs vector Hj(w), i G N, from the y-S-P Core of the original game rx°. This property means that a cooperative

yy

could be implemented as a subgame-perfect equilibrium) in a closely related non-cooperative game r^0.

y

of an extensive-form game rx° G (n) is non-empty, and y = yi(xt), i G N, Xt G w y

ists a subgame-perfect equilibria u = (ui,..., un) a related non-cooperative game rX° which generates a cooperative history w = (x0,...,XT) with a SPE payoffs

T

vector ^¿(u) = yi(xt) = yi(w), i G N.

t=o

A refinement of Nash equilibria concept called Strong NE (SNE) was introduced in (Aumann, 1976) for strategic games. As it turns out, if an extensive game r C (n) admits a unique subgame perfect SNE then y-S-P Core consists of the unique simplest payoff distribution procedure.

Definition 7. (Chander and Wooders, 2020) A pure strategy profile u = (ui, ..., un) is a subgame perfect SNE of rx° C (n) if uisa SPE of each induced gamer x°'s, S C N.

If an extensive game possesses subgame perfect SNE u then no coalition has a profitable joint deviation from u in any subgame.

Note that a pure strategy definition for extensive game originally implies some redundancy (see, for instance, (Kuhn, 1953; Myerson, 1997; Petrosyan and Kuzyutin, 2000) for details), namely two strategy profiles could define different choices of some players at some nodes but generate the same path in the game tree and the same players' payoffs. In what follows, we will refer to "a game r possesses a unique SPE or r has multiple SPE but all of them generate the same path in the game tree" as "a game r admits a unique SPE".

Proposition 2. Let each induced game rx°'s, S C N, of the game rx° C (n) admits a unique SPE. If rx° admits a subgame perfect SNE then this subgame

y

procedure, namely:

yi(xt) = hi(xt), xt G w.

(12)

Proof. The proof of the first statement is straightforward and has been already provided in (Chander and Wooders, 2020). Let u = (ui,... , un) denote the unique subgame perfect SNE in rX0.

Since u is a unique SPE in rX°'N it generates exactly the cooperative history w = (x0,...,xT) in the game tree. Moreover, a unique SPE at each induced game rX°,S, S c N, is u and every induced subgame rXt'S, S c N, t = 0,..., T - 1 admits a unique SPE (namely, the restriction of u onto the subgame rXt'S) that generates a SPE history wXt = (xt,..., xT). Then (10) takes the form

T T

Y (S; xt ) = ^ hs (xT) < ^ fts (xT), t = 0,...,T - 1. (13)

T=t T=t

Taking (5), (6) and the strict balance condition (7) into account we obtain from (13) by backwards induction that

fti(xt) = hj(xt), xt G w.

ftft

between the players while the game unfolds along the cooperative history w. □

Corollary 1. If a two-player extensive-form game rX° G (2) admits a unique

ft

ft

cept than the subgame-perfect SNE. For instance, the extensive game from Example

ft

6. Refinement of the ,3-S-P Core

ft

of multiple payoff distributions procedures.

ft

form game r G (n) is non-empty then it is a (closed) con vex polytope B in

Rnx(T +i)_

It is worth noting that the main purpose of each player j when selecting a unique PDP ft from ft-S-P Core is the value pj = ftj(w) which the j-th player will get according to the cooperative agreement, given that distribution {ftj(xt)}; xt G w, of the /^¿(w) along the cooperative path satisfies additional constraints (6), (8) and (10).

ft

ment, i.e., some rules for selecting ftj(w), j G N. One approach is to maximize the

j

ftj - Y({j}; xo) ^ max . (14)

Note that (14) is a linear programming problem which has a solution due to Prop. 3. If one apply this approach to the game in Ex.2 assuming that the 2-nd player's payoff ft2 is the goal function the resulting PDP will be ft' : fti = 14, 32 = 10.

Another approach is to adopt some bargaining solution (see, e.g. Moulin, 1988; Castanr et al., 2021) to choose a unique vector (/j)jeN from B when the subgame perfect equilibrium payoffs vector (Y({j}; xo))jeN serves as a disagreement point (point of the "status quo"). For instance, when employing the symmetric Nash bargaining solution, the corresponding problem takes the form

n(A - Y({i}; xo)) ^ max . (15)

If we apply approach (15) to the game in Ex.2 we should choose PDF / : ¡31 =

15, /2 = 9.

Let us focus in the paper on the third approach (introduced in (Kuzyutin and Smirnova, 2021)) which takes care of the relative benefit from cooperation (RBC) of the least winning player. Let Aj denote the absolute range

of the j-th player's payoffs in r^ -Y({j}>Xo) could be interpreted as the relative benefit of player j from cooperation according to the PDF Then, the players are expected to find the solution of the following optimization problem

• - Y({j}; xo) maxmm —--. (16)

/eB jew Aj

Remark 3. For two-player game problem (16) takes the form

A - y({1}; xo) = /2 - Y({2}; xo) Ai A2 '

Let us illustrate the latter approach using 2-person game in Ex. 2: ^ - 14 /2 - 8 (15 + e) - 14 (9 - e) - 8

(17)

1^ — 6 1^ — 6 10 6

e = 0.2.

Hence, /31 = 15.2, /2 = 8.8 while the exact distribution of /j along the cooperative history w satisfies constraints (7), (8) and (10). For instance, the following PDF /3" meets all the constraints.

w xo X2 X6 X7 j(w)

/i'(xt) 4.2 1 2 8 15.2

№t) 1.8 4 2 1 8.8

We will refer later to the latter approach for the /3-S-P Core refinement as the maxmin RBC rule. Note that different approach to the evaluation of the relative benefit from cooperation (as well as for the "value of the preexisting knowledge") was proposed in (Chebotareva et al., 2021) for differential games.

Another way to contract the /3-S-P Core is to assume that a PDF /3 from the Core has to satisfy some additional properties. For instance, let us consider the irrational-behavior-proof property for cooperative solution that is implemented via IDF /3 in an extensive-form game (this property was firstly introduced in (Yeung, 2006) for cooperative differential games).

Definition 8. (Yeung, 2006; Kuzyutin and Nikitina, 2017; Kuzyutin et al., 2019b) The PDP ft for the payoff vector (pi;... ,pn) meeting (5), (6) satisfies irrational-behavior-proof (IBP) property (for coalitions) if at each node xt G w = (xo,..., xT), t = 1,..., T - 1, for all coalitions S which are active at xt the following inequality-holds:

t-i

^fts(xt) + y(s; xt) > y(s; xo). (18)

T=o

S

xo xt

S xt

"irrational behavior" of some other players.

S

payoff given the following mode of partial cooperation: the players cooperate along

xo xt

(namely - SPE) behavior.

ft ft'

ft

ft' S = {1}

x2

fti(xo) + Y({1}; x2) = 4 + 10 > 14 = y({1}; xo),

fti(xo) + Y({1}; x2) = 3 + 10 > 14 = y({1}; xo). ft

ft

7. /3-S-P Core for fishery-management model

ft

renewable resource extraction (see, e.g., Levhari and Mirman, 1980; Mazalov and Rettiyeva, 2011; Breton et al., 2019; Chander, 2017; Ougolnitsky and Usov, 2019; Mazalov et al., 2021) we use a 2-player fishery-management model in extensive form introduced in (Kuzyutin and Smirnova, 2021) which is a finite version of the original fishery-management model (Levhari and Mirman, 1980) that has been studied in (Haurie et al., 2012). We suppose in the paper that competing countries equally evaluate the worth of the resource (fish biomass) remainder after the resource extraction (fishery) process ends (namely, Ki = K2 = 1), whereas in (Kuzyutin and Smirnova, 2021) the players differently appreciate the resource remainder (the reasons for asymmetric environmental valuation were discussed, e.g.,

in (Cabo and Tidball, 2021)). Note, that cooperative path and payoff distribution

ft

sentially differ from the cooperative solution that was obtained in (Kuzyutin and Smirnova, 2021) for the case of asymmetric environmental valuation.

Example 3. (A 2-player symmetric fishery-management model in extensive form).

Let y(t) denote a (normalized) fish amount in year t, t = 0,1,..., T, that evolves according to the equation

y(t +1) = a • y(t),

where a > 1 denotes the annual net growth rate. Assume that two players (e. g., companies or countries) exploit the fishery and let uj (t) > 0 denote the harvest of player j in year t. Given the initial condition y(0) = yo the system dynamics is described by the state equation

y(t + 1) = a • (y(t) - (ui(t) + u2(t))), (19)

where 0 < ui(t) + u2(t) < y(t).

j

t-i _

Hj(•) = E ¿j^j) + Kj • j^M, j = 1, 2, (20)

t=o

where ¿j G [0,1) is the jth player's discount factor while Kj > 0 describes how the j

period).

Some additional assumptions are accepted in (Kuzyutin and Smirnova, 2021) to embed the fishery-management model (19)-(20) into an extensive-form game framework. Firstly, suppose that both players can fish out at only two levels: Low (uL = Lj) or High (uH = Hj) and consider two-year model, i. e., T = 2. Further, to obtain a game with perfect information we suppose that in every year player 1 moves (i.e. chooses a particular level of ui) first.

The resulting discrete-time symmetric fishery-management model in extensive form for particular parameter values: y(0) = 10 a = 1.25; uH = Hj = 3, uL = Lj = 1; ¿i = ¿2 = £ = 0.9; Ki = K2 = 1, is presented in figure 2. Note that Pi = {xo,x3,x4,x5,x6}, P2 = {xi,x2,xr - xi4}, P„+i = {zi,..., zi6}, the right alternative at every node xk G Pj corresponds to high level of fishery efforts (harvest) (uj(xk) = Hj). It is worth noting that the current payoffs -y/uj (0) in xi - x6 correspond to the first period of fishery process (i. e., ¿o = 1 according to (20)) whereas the current payoffs ^/uj (1) in x7 - xi^d zi - zi6 correspond to the second year (i. e., ¿i = 0.9). The payoifs hj(xo), j = 1, 2, at the origin xo could be considered as the players' initial assets.

The extensive-form game described above possesses a unique SPE (the corresponding equilibrium choices at every decision node are presented as dotted lines in fig. 2). This SPE generates the path wSPE = (xo, x2, x5, xi2, zi2) with the resulting non-cooperative payoffs (5.4; 4.67).

The cooperative path w = (xo,xi,x3,x7, zi), which is emphasized in bold in fig. 2, corresponds to the highest fish biomass remainder (both environmentally-concerned players each year fish out at low level) and implies maximal summary-payoff hi(w)+ h2(w) = 5.46 + 5.46 = 10.92.

Note that the system (5)-(8) and (10) for the extensive-form fishery-management

ft

(ft^w),ft2(w)) from the ft-S-P Core we employ the maxmin RBC rule. Using the notations ftSi(w) = 5.46 + e, ft2(w) = 5.46 - e, we obtain equation (17) in the form

(5.46 + e) - 5.4 (5.46 - e) - 4.67 -—- < > e — 0.3.

5.98 - 4.54 5.98 - 4.01

Hence, A(w) = 5.76, ft2(w) = 5.16. The table below presents an example of particular PDP /3 from the ft-S-P Core that was determined via backwards induction and

0.9 1.56 0.9 1.56 0.9 1.56 0.9 1.56 0.9 1.56 0.9 1.56 0.9 1.56 0.9 0

Value of tlie fisli biomass remainder ß2\/y(2) : 2.56 2.22 2.22 1.81 2.12 1.69 1.69 1.11 2.12 1.69 1.69 1.11 1.57 0.91 0.91 1.28

Total players' payoffs (20) :

5.46 5.12 5.78 5.37 5.02 4.59 5.25 4.67 5.75 5.32 5.98 5.4 5.2 4.54 5.2 5.57

5.46 5.78 5.12 5.37 5.75 5.98 5.32 5.40 5.02 5.25 4.59 4.67 5.2 5.2 4.54 4.01

Fig. 2. Fishery-management symmetric 2-player model in extensive form

implies no payoff transfers at terminal node z1 and minimal transfers between the players at decision nodes in the cooperative path.

<jJ xo Xl X3 X7 zi ß (*)

ßl(xt) 1.39 1 0.23 0.58 2.56 5.76

ß2(xt) 0.61 0 0.77 0.32 3.46 5.16

ßi(xt) - hi(xt) 0.39 0 0.23 -0.32 0 0.3

Note that the exact values of the payoff transfers (from player 2 to player 1) which are necessary at each node of the cooperative path to ensure the sustainability of the cooperative agreement are given in the lowest table row.

8. Conclusion

The paper contributes to the theory of cooperative behavior in extensive-form games. Firstly, we compare the subgame-perfect core (Chander and Wooders, 2020) and /3-S-P Core (Kuzyutin and Smirnova, 2021) and demonstrate in Sections 3 and 4 that the latter solution is more powerful tool to implement a cooperative agreement (this implication was briefly noted in (Kuzyutin and Smirnova, 2021) without discussion). Secondarily, in Section 5 we reveal relationship between strong Nash equilibria and /3-S-P Core for the class of extensive-form games with perfect information and payoffs defined at each node. Lastly, in Section 6 we propose several

approaches for the /3-S-P Core refinement (some of them require an optimization problem solution, others imply that particular PDP from the /3-S-P Core has to satisfy additional properties like irrational-behavior-proof condition) and illustrate these approaches using two examples.

Note that /3-S-P Core approach is very promising and could be applied to other classes of extensive-form games (e.g., extensive games with chance moves (Kuzyutin and Smirnova, 2020) or extensive-form multicriteria games (Kuzyutin and Nikitina, 2017; Kuzyutin et al., 2018; Kuzyutin et al., 2019b). It is also of interest how one can adapt the /3-S-P Core concept in the discrete-time models of dynamic interaction between several religious and political movements (see, e.g., Tantlevskij et al., 2021; Kuzyutin et al., 2020).

Acknowlegments. The research of the first author was funded by RFBR, project number 21-011-44058.

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