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Replicator dynamics with Pigovian subsidy and capitation tax
Nonlinear Analysis 71 (2009) e818–e826
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Replicator dynamics with Pigovian subsidy and capitation tax Takafumi Kanazawa, Yasuhiko Fukumoto, Toshimitsu Ushio ∗ , Takurou Misaka Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan
article Keywords: Evolutionary game Replicator dynamics Tax and subsidy
info
abstract The selfish behaviors of individuals can cause an inefficient society in which the total amount of the individuals’ utilities is not maximized. To resolve the problem, a government sometimes tries to control the population by imposing a tax on and/or offering a subsidy to the individuals who belong to the population. The tax is roughly classified into rate taxations and capitation taxes. Using an evolutionary game theory, the authors have proposed a differential equation model with rate taxations to analyze their effects on players’ behaviors. In this paper, we propose a differential equation model with the capitation taxes, and derive stability conditions of the target state. Moreover, we also discuss an application of our model to selfish routing games. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In real world, there exist many populations which consist of a large number of selfish players interacting with each other. In such a population, the purpose of each player often conflicts with the total purpose of the population, and a problem called a social dilemma occurs [1]. Evolutionary game theory has been used as a powerful mathematical framework to analyze social dilemma [2], and the dilemma game is characterized by replicator dynamics in Ref. [3]. Such a conflict is caused by several exogenous factors. Recently, a replicator equation on graphs [4] and a model of active linking [5] have been proposed to analyze effects of players’ interaction structures which is one of the exogenous factors in evolutionary games. In the replicator equation on a regular graph, each node represents a player and each edge means that players corresponding to its nodes can play a game. In the model of active linking, players form interaction links which have specific life-times at different rates and their dynamics are proposed. In these models, the interaction structures are modeled by transformations of the payoff matrices, and their effects on evolutions of cooperations are discussed [4,5]. To resolve the conflict between each player and the population, another exogenous factor called externality has to be taken into account. The word externality used in economics is an effect of a benefit obtained without paying payable costs or a cost paid without obtaining a receivable benefit [6]. Since the social dilemmas involve externalities, they cannot be resolved by the personal effort of each player. A government which has the comprehensive perspective is required for governing the population. In real world, it corresponds to the rulers such as governments of countries or cities, and executives of organizations or companies. To correct the externalities, a government sometimes tries to control the population by imposing a tax on and/or offering a subsidy to players who belong to the population. Such taxes and subsidies are called Pigovian taxes and Pigovian subsidies, respectively [6]. The tax is roughly classified into rate taxations and capitation taxes. In the former, the tax is determined based on payoffs the players earn while it is fixed in the latter. Using an evolutionary game theory, the authors have proposed a differential equation model with rate taxations to analyze the effects of rate taxations on players’ behaviors [7]. In this model, the government is willing to lead the population state to a desirable target state by using rate taxations and subsidies which
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depend on the target state. The amount of taxes which are imposed on each player is proportional to its payoff in the rate taxation model. Different from the rate taxation, the capitation taxes are determined independent of the players’ payoffs and the same amount of taxes is imposed on each player. In this paper, we deal with the capitation tax. To analyze effects of capitation taxes on players’ behaviors, we propose replicator dynamics with capitation taxes and subsidies. In this model, the government is willing to control the population state to a desirable target population state by collecting capitation taxes from all players and reallocating them as subsidies depending on the target state. We derive several properties of our model and provide the capitation tax which can make the target state locally or globally asymptotically stable. Moreover, we also discuss an application of our model to selfish routing games. 2. Preliminaries Consider a population which consists of a large number of players. Each player adopts a strategy which is inherited from its parent. In evolutionary games, repeatedly, two players are randomly drawn from the population to play a game with, and players produce offspring whose numbers depend on payoffs of the game [8]. In the context of social systems, this selection process is perceived as a model where players change their strategies according to the payoffs which depends on the results of the games. In this paper, we use the following notations: S = {1, . . . , n}: the set of players’ pure strategies; ei ∈ Rn (i = 1, . . . , n): a unit vector which corresponds to a pure strategy i; ∆ ⊂ Rn : the mixed strategy space (a convex combination of ei for all i ∈ S); int(∆): the interior of the mixed strategy space ∆; bd(∆): the boundary of the mixed strategy space ∆; C (x)(:= {i ∈ S |xi > 0}) ⊆ S: the carrier of the mixed strategy x; x = (x1 , . . . , xn )T ∈ ∆: a population state (a distribution of strategies, where xi is a proportion of players with strategy i); • A = (aij ) ∈ Rn×n : the payoff matrix; and • u(x, y) := xT Ay: the payoff function of players with mixed strategy x against y.
• • • • • • •
From the above definitions, conventional replicator dynamics is formulated as: x˙ i = {u(ei , x) − u(x, x)}xi
(1)
for each strategy i ∈ S. 3. Rate taxations We consider that the government collects a part of players’ payoffs at a taxation rate α ∈ [0, 1] and reallocating them as subsidies depending on the target state. If α = 0, then the government takes no taxes from the players. Such a situation is called non-intervention. On the other hand, if α = 1, then all payoffs which players earn are collected and reallocated by the government. Such a situation is called complete intervention. Suppose P that the number of players who adopt a strategy i ∈ S is pi > 0 and the number of all players in the population is p = i pi . If the government imposes the tax at rate of α on P each player, then the total payoff which is collected from players is α i pi u(ei , x). Denoted by x∗ ∈ ∆ is a target state which the government considers as a desirable population state. Suppose that the more desirable strategy for the government players adopt, the P more subsidies they are provided. Then, P thePsubsidy which is provided to players with a strategy i is assumed to be x∗i α j pj u(ej , x). Note that a summation i x∗i α j pj u(ej , x) of the reallocated payoffs to all players is equal P to the total tax which is collected from players since i x∗i = 1 holds. The reallocated subsidy to each strategy is equallydivided P to players with the strategy. P Therefore, the subsidy which is provided to each player with a strategy i is given by α x∗i j pj u(ej , x)/pi = α(x∗i /xi ) j∈S xj u(ej , x). Thus, a payoff function for the players with rate taxations and subsidies is given by the following function:
(1 − α)u(ei , x) + α
x∗i xi
u(x, x).
(2)
The first term of Eq. (2) is uncollected part of payoffs and the second one is collected and reallocated part of payoffs. Note that, since we assume that all collected taxes are reallocated in this case, the average payoff of players u(x, x) is independent of the taxation rate α . Moreover, suppose that every pure strategy is adopted by some players at least in the initial state, that is, xi (0) > 0 is assumed to hold for any pure strategy i ∈ S. In other words, we assume that x(0) ∈ int(∆). By this assumption, within any finite-time interval, xi > 0 holds for all i ∈ S and Eq. (2) is well defined. In this model, we assume that u(ei , x) ≥ 0 holds for any i ∈ S, and there exists j ∈ S satisfying u(ej , x) > 0 for all x ∈ ∆. Under this assumption, u(x, x) > 0 holds for any x ∈ ∆. Substituting Eq. (2) for players’ payoff function u(ei , x) of Eq. (1), we have replicator dynamics with rate taxations and subsidies as the following equation: for each strategy i ∈ S, x˙ i = (1 − α) xi {u(ei , x) − u(x, x)} + α x∗i − xi u(x, x),
(3)
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where α ∈ [0, 1] is the taxation rate. Taxation rates which can make the target state locally or globally asymptotically stable has been proposed in Ref. [7]. We summarize properties of the rate taxation model. By Eq. (3), obviously, if the government adopts the strategy non-intervention (α = 0), then Eq. (3) is reduced to Eq. (1). On the other hand, it has been proved if the government adopts the strategy complete intervention (α = 1), then the target state x∗ ∈ ∆ is a globally asymptotically stable equilibrium point. For the target state x∗ , we obtain the following proposition [7]: Proposition 1. If the target state x∗ ∈ ∆ is an equilibrium point of Eq. (1), then it is an equilibrium point of Eq. (3) for any α ∈ [0, 1]. On the other hand, if x∗ is not an equilibrium point of Eq. (1), then there does not exist α ∈ [0, 1) such that it is an equilibrium point of Eq. (3). Proposition 1 means if x∗ is not an equilibrium point of Eq. (1), then the government has to adopt complete intervention for leading the population state to the target state. Therefore, we consider the case that the target state is an equilibrium point of Eq. (1). We define a subset W ⊆ ∆, a scaled space Uε ⊆ ∆ of ∆ with the scaling factor ε ∈ (0, 1] and the center x∗ , and a function α( ˜ x) of x ∈ W as follows:
) X ∗ W = x ∈ ∆ x u(ei , x) < u(x, x) , i∈S i Uε = x¯ ∈ ∆ x¯ = (1 − ε)x∗ + ε x, x ∈ ∆ , P ∗ u(x, x) − xi u(ei , x) i∈S o . α( ˜ x) = P n x∗ ∗ xi xi u(x, x) − u(ei , x) i (
(4) (5)
(6)
i∈S
The target state x∗ is a locally asymptotically stable equilibrium point of Eq. (3) for all α ∈ (α, 1], where α = 0 if there exists ε > 0 satisfying Uε ∩ W = ∅, otherwise α = supx∈Uε ∩W α( ˜ x) for some ε > 0 satisfying Uε ∩ W 6= ∅. Since Uε equals to ∆ with ε = 1, we obtain the following theorem [7]: Theorem 1. We set α = 0 if W = ∅ and α = supx∈W α( ˜ x) if W 6= ∅. Then, x∗ is a globally asymptotically stable equilibrium point of Eq. (3) for all α ∈ (α, 1]. 4. Capitation taxes 4.1. Model In this section, we deal with the capitation taxes. Different from the rate taxation, the capitation taxes are determined independent of the players’ payoffs. To analyze their effects on players’ behaviors, we propose replicator dynamics with capitation taxes and subsidies. Suppose P that the number of players who adopt a strategy i ∈ S is pi > 0 and the number of all players in the population is p = i pi . If the government imposes the capitation tax c on each player, then the total payoff which is collected from players is cp. Let x∗ be a target state. Suppose that the more desirable strategy for the government players adopt, the more subsidies they are provided. Then, the subsidy which is provided to players with a strategy i is assumed to be cpx∗i . The reallocated subsidy to each strategy is equally-divided to players with the strategy. Therefore, the subsidy which is provided to each player with a strategy i is given by cpx∗i /pi = cx∗i /xi . Thus, the payoff function for the players with subsidies and capitation taxes is given by the following function: u(ei , x) − c + c
x∗i xi
= u(ei , x) − c 1 −
x∗i xi
.
(7)
Similar to the model of the rate taxations, we assume that all collected taxes are reallocated and the average payoff of players is independent of the capitation tax c. We assume that x(0) ∈ int(∆) and Eq. (7) is well defined within any finitetime interval. Substituting the right-hand side of Eq. (7) for players’ payoff function u(ei , x) of Eq. (1), we have replicator dynamics with subsidies and capitation taxes as follows: for each strategy i ∈ S, x˙ i = {u(ei , x) − u(x, x)}xi + c (x∗i − xi ),
(8)
where c ≥ 0 is the amount of the capitation taxes. Note that we allow the case that the tax c is greater than the players’ original payoffs, that is, we allow c > u(ei , x). Eq. (8) is given by adding the negative feedback term c (x∗i − xi ) to the conventional replicator dynamics equation (1), and the tax c is considered as a feedback gain.
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4.2. Properties We show several properties of Eq. (8) and prove a condition for the capitation tax to make the target state locally or globally asymptotically stable. See Appendix for proofs of propositions, theorems, and a corollary in this section. Proposition 2 (Invariance Under a Local Shift). Eq. (8) is invariant under a local shift of the payoff matrix A, where the local shift is the addition of a constant to all elements of a column of A. Eq. (1) is invariant under the local shift of the payoff matrix A, but the rate taxation model proposed by [7] is changed by the local shift. Using the invariance under the local shift, without loss of generality, we can suppose that each element of the payoff matrix A is non-negative. Especially, in two-strategy game, we can simplify the matrix A by setting all non-diagonal elements to zero. Proposition 3 (Equilibrium Target Point). If the target state x∗ is an equilibrium point of Eq. (1), then it is an equilibrium point of Eq. (8) for any tax c > 0. Generally, it is not always true that the target state x∗ is an equilibrium point of Eq. (8). However, if the target state x∗ is an equilibrium point of Eq. (1), then x∗ is always an equilibrium point of Eq. (8). In this paper, we focus on the case that the target state x∗ is an equilibrium point of Eq. (1), and discuss its stability. Theorem 2 (Locally Asymptotic Stability). Let the linearization system of Eq. (1) at the target state x = x∗ be x˙ = J0 x, and the eigenvalues of the Jacobian matrix J0 of Eq. (1) at x = x∗ be λ0i (i = 1, . . . , n). Then, the linearization system of Eq. (8) at the target state x = x∗ is given by x˙ = (J0 − cIn )x,
(9)
where In is the n × n unit matrix. The origin is asymptotically stable in Eq. (9) if and only if c > maxi {R(λ0i )}, where R(λ0i ) is the real part of λ0i . If the origin of the linearization system equation (9) is asymptotically stable, then the target state x = x∗ of the nonlinear system equation (8) is locally asymptotically stable. In the case that c = maxi {R(λ0i )} holds, although the origin of Eq. (9) is Lyapunov stable, the stability of x = x∗ depends on the higher-order terms of Eq. (8). Theorem 3 (Globally Asymptotic Stability). Suppose that each element of the payoff matrix A is non-negative and define c¯ (x) as follows: for all x ∈ int(∆) \ {x∗ }, n P
c¯ (x) =
(xi − x∗i )u(ei , x)
i=1 n P i=1
x∗
.
(10)
(x∗i − xi ) xii
If the target state x∗ is a Nash equilibrium, then it is a globally asymptotically stable equilibrium point of (8) for c > max{0, supx∈int(∆)\{x∗ } c¯ (x)}. Moreover, c¯ (x) < 2(n − 1) maxi,j aij for any x ∈ int(∆) \ {x∗ }. Due to Proposition 2, for a game with a payoff matrix A˜ which has a negative element, we transform it to a payoff matrix A whose elements are non-negative by the local shift to the matrix A˜ and apply Theorem 3 to the matrix A. The target state of the game with A˜ is also stabilized globally by the tax c for the game with A. Now, we intervene a population state x as a situation that all players of the population adopt a mixed strategy x. Then, the numerator of c¯ (x) is a difference between a payoff of a player with a mixed strategy x and that of a player with x∗ . c¯ (x) is positive if the former is larger than the latter while it is negative if the latter is larger than the former. Since players have no incentive to change their strategy x to x∗ if the former is larger, the achievement of the target state x∗ requires the government’s intervention. The larger the difference is, the larger the tax c which can stabilize the target state x∗ is. On the other hand, if the latter is larger, then players are willing to change their strategy x to x∗ independent of the tax c. The denominator of c¯ (x) estimates a kind of distance between a population state x and the target state x∗ . Since the distance is not a simple summation of differences between xi and x∗i but a weighted summation of the differences multiplied by x∗i /xi , the positive differences are amplified and the negative differences are discounted. So, roughly speaking, the farther a population state x is from the target state x∗ , the larger the denominator is. To set c = 2(n − 1) maxi,j aij as a tax means that the government imposes the larger tax on players when the number of strategies and the maximum payoff are larger. Theorem 3 supposes that the target state x∗ is a Nash equilibrium. The target state which is not a Nash equilibrium must be on the boundary of ∆, that is, x∗ ∈ bd(∆). For such a target state, we have the following theorem: Theorem 4. Suppose that each element of the payoff matrix A is non-negative and x∗ ∈ bd(∆). For all i 6∈ C (x∗ ) and for any c > maxi6∈C (x∗ ) maxj∈S aij , orbits x(t ) of Eq. (8) satisfy limt →∞ xi (t ) = 0.
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Fig. 1. Phase portrait for c = 0.
When x∗i = 0 and xi (0) = 0, the set {x ∈ ∆|xi = 0} ⊂ bd(∆) is a positive invariant set since x˙ i (t ) ≡ 0. Therefore, it is expected that we can find the tax c which makes the target state an attractor of Eq. (8) by the following procedure. First, using Theorem 4, find a tax c1 which converges xi to zero for all strategies i 6∈ C (x∗ ). Next, for a sub-game which omits strategies i 6∈ C (x∗ ), find a tax c2 which stabilizes the target state x∗ globally using Theorem 3. Finally, we set c = max{c1 , c2 } as a tax so that the target state x∗ becomes an attractor of Eq. (8) whose basin is ∆. However, it is not obvious that the tax c which is given by the above procedure makes the target state x∗ ∈ bd(∆) an attractor of Eq. (8) since the orbits of Eq. (8) cannot converge to the set {x ∈ ∆|xi = 0} ⊂ bd(∆) within finite-time intervals. Even if the tax can make the target state an attractor, the target state may not be Lyapunov stable. When the target state x∗ is a vertex of ∆, we can prove x∗ is not only an attractor but also a Lyapunov stable equilibrium point, that is, x∗ is a globally asymptotically stable equilibrium point. Corollary 1 (Stability of the Target State on a Vertex). Suppose that the target state x∗ is a vertex of ∆. Then it is a globally asymptotically stable equilibrium point of Eq. (8) if c > maxi∈S \{k} maxj∈S aij . 4.3. Example Heavy traffic in urban areas cause several kinds of problems such as air pollution, increase in CO2 level, and so on. Recently, park and ride (P&R) is paid much attention to as a useful system for a traffic congestion problem. Park and ride means that people who wants to go the urban areas drive their cars by a train station or a bus stop, park their cars there, and transfer to a public transport system such as a train or a bus. As the other option to reduce traffic congestion, they use bicycles. In this section, suppose that the strategy 1, 2, and 3 correspond to go by ‘‘Car’’, ‘‘P&R’’, and ‘‘Bicycle’’, respectively. We set players’ payoff matrix A as follows: 1 7 6
" A=
8 2 5
9 4 . 3
#
(11)
Fig. 1 shows its phase portrait. There exists a unique stable equilibrium point ◦, three unstable equilibrium points •, and three saddle points in Fig. 1. In the case that the government does not impose taxes, all orbits whose initial states are in int(∆) converge to the unique interior stable equilibrium point. Suppose that the government perceives the case that no one uses his/her car is the desirable population state, that is, we consider the saddle point (0, 1/4, 3/4) as the target state x∗ . Consider the linearization system at the target state x∗ . Note that x always satisfies x1 + x2 + x3 = 1. We can eliminate x3 by substituting 1 − x1 − x2 for x3 . Since the eigenvalues of the Jacobian matrix at x∗ are 21/4 and −3/4, Theorem 2 shows that the target state is a locally asymptotically stable equilibrium point if c > 21/4. Figs. 2 and 3 show transient controlled behaviors with c = 26/5 (<21/4) and c = 53/10 (>21/4), respectively. In the case of c = 26/5, all orbits converge to a neighborhood of the target state, but do not converge to the target state itself as shown in Fig. 2. However, in the case of c = 53/10, all orbits converge to the target state as shown in Fig. 3. Theorem 4 shows that all orbits converge to a surface with x1 = 0 where the target state x∗ ∈ bd(∆) is if c > 9. A two-strategy sub-game which omits the strategy 1 has a unique locally asymptotically stable equilibrium point (x2 , x3 ) = (1/4, 3/4) if c = 0. Theorem 3 shows that the target state is globally stabilized if c > 0. Depicted in Fig. 4 is a phase portrait for c = 91/10, which shows that the target state is globally asymptotically stable. 5. An application of capitation tax model to selfish routing A selfish routing game is a simple model of selfish behaviors in networks [9,10]. Its replicator dynamics has been proposed [11]. In such a game, there exists inefficiency caused by selfish route selections of each player and several
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Fig. 2. Transient behavior of x1 for c = 26/5.
Fig. 3. Transient behavior of x1 for c = 53/10.
Fig. 4. Phase portrait for c = 91/10.
methodologies have been proposed to reduce it. In this section, we consider an application of the capitation tax model to such a selfish routing game. In selfish routing games, suppose that there is a fixed flow demand and it is routed from a source to a sink. A player’s strategy is a selection from all possible paths from the source to the sink, and its payoff is related to latency on each path. Since the player selects a path in order to minimize the latency, the payoff of selecting a path i ∈ S is set to u(ei , x) = −li (x), where li (x) is latency of a path i with flow x ∈ ∆. We can consider the capitation tax c in our proposed model is additional fictitious delay to flow of each path in routing games. On the other hand, we cannot introduce the subsidies into the selfish routing since we cannot reduce latency in network by the control. Therefore, using Proposition 2, we modify our proposed payoff function with subsidies and capitation taxes given by Eq. (7) as follows:
u(ei , x) − max c i∈S
x∗i xi
+c
x∗i xi
.
(12)
Substituting Eq. (12) for player’s payoff function u(ei , x) of Eq. (1), we have Eq. (7). Since Eq. (12) is always less than or equal to u(ei , x) for all i ∈ S with equality if and only if x = x∗ , it is well defined as players’ payoff functions in selfish routing games.
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Fig. 5. Braess’ Paradox. Each edge is labeled with its latency, where xei is the amount of traffic using the edge which is labeled with it, that is, xe1 = x1 + x2 and xe2 = x2 + x3 .
Fig. 6. Phase portrait for c = 0.
As a simple example, we consider a network shown in Fig. 5. Suppose that there is a fixed flow demand 1 and it is routed from source s to sink t. This network has three s–t paths. Let strategies 1, 2, and 3 correspond to a selection of the paths s–v –t, s–v –w –t, and s–w –t, respectively. The payoff matrix is given by latency of each edge of the network shown in Fig. 5. By a local shift of the given payoff matrix, we have the following payoff matrix whose elements are non-negative: 0 1 1
" A=
0 0 0
1 1 . 0
#
(13)
In this network, a Nash equilibrium flow is x = (0, 1, 0), while the minimum latency flow is (1/2, 0, 1/2). Such a situation is well known as Braess’s Paradox. We consider the minimum latency flow (1/2, 0, 1/2) as the target state. Consider the linearization system at the target state x∗ . The eigenvalues of the Jacobian matrix at x∗ are 1/2 and −1/2. Theorem 2 shows that the target state is a locally asymptotically stable equilibrium point if c > 1/2. Moreover, Theorem 4 shows that all orbits converge to a surface with x2 = 0 if c > 1. A two-strategy sub-game which omits the strategy 2 has a unique locally asymptotically stable equilibrium point (x1 , x3 ) = (1/2, 1/2) for any c ≥ 0, and Theorem 3 shows that the target state is globally stabilized if c > 0. Figs. 6 and 7 show phase portraits of the selfish routing with taxes and subsidies for c = 0 and c = 11/20 (>1/2), respectively. In this example, by Theorem 2, a tax satisfying the condition c > 1/2 stabilizes the target state globally as shown in Fig. 7. By Theorem 4, all orbits converge to a surface with x2 = 0 if a tax satisfies the condition c > 1. 6. Conclusions In this paper, we have proposed replicator dynamics with capitation taxes and subsidies to analyze effects of capitation taxes on players’ behaviors. In this model, the government’s purpose is to control the population state to a target state by collecting capitation taxes and reallocating them as subsidies depending on the target state. We have proved several properties of our model and have provided a capitation tax which can make the target state locally or globally asymptotically stable. Moreover, we have also discussed an application of our model to selfish routing games. In our model, the amount of taxes is assumed to be independent of the population state. However, we can consider that it is changed depending on the population state as a government’s policy. It is our future work to propose a model which can deal with such changes of capitation taxes.
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Fig. 7. Phase portrait for c = 11/20.
Acknowledgments This research was supported in part by KAKENHI (No. 19860045) and ‘‘Global COE (Centers of Excellence) Program’’ of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Appendix. Proofs A.1. Proof of Theorem 2 Let the linearization system of Eqs. (1) and (8) at the target state x = x∗ be x˙ = J1 (c )x and x˙ = J0 x, respectively. Moreover, we denote Eqs. (1) and (8) by x˙ i = fi (x) and x˙ i = fi0 (x), respectively. Obviously, x˙ i = fi0 (x) = fi (x) + c (x∗i − xi ) holds. Then, we have
∂f 0 1
···
∂ x1 . J1 (c ) = .. ∂f 0 n
..
.
···
∂ x1
∂f
∂ f10 ∂ xn .. . 0 ∂f
1
∂ x1 .. = . ∂f
n
∂ xn
−c
··· ..
n
···
∂ x1
x=x∗
.
∂ f1 ∂ xn .. .
∂ fn −c ∂ xn
,
(A.1)
x =x ∗
that is, J1 (c ) = J0 − cIn . Therefore, if we suppose that the eigenvalues of J1 (c ) are λ11 , . . . , λ1n , then λ1i = λ0i − c holds for i = 1, . . . , n. Note that λ0i is a constant which depends on the payoff matrix A and the target state x∗ . Thus, the real parts of all eigenvalues of J1 (c ) are negative, that is, the origin of Eq. (9) is an asymptotically stable equilibrium point if and only if c > maxi {R(λ0i )} holds. A.2. Proof of Theorem 3 Consider the following function: V (x) =
X
∗
−xi log
i x∗ >0 i
xi x∗i
.
(A.2)
This function satisfies V (x) ≥ 0 with equality if and only if x = x∗ . The time derivative of V (x) along solutions of Eq. (8) is V˙ (x) = −
X
(x∗i − xi )u(ei , x) −
i
X x∗ (x∗i − xi ) i c . i
xi
(A.3)
Then V˙ (x) < 0 for all x 6= x∗ if
( c > max 0,
) sup
x∈int(∆)\{x∗ }
(¯c (x)) .
Thus, x∗ is a globally asymptotically stable equilibrium point.
(A.4)
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We will prove that 2(n − 1) maxi,j (aij ) is an upper bound of c¯ (x). Let δ = (δ1 , . . . , δn ) be a difference between x and x∗ , that is, δi = xki − x∗ki for i = 1, . . . , n. Without loss of generality, we assume that δi > 0 for i = 1, . . . , np , δi < 0 for i = np + 1, . . . , np + nm , and δi = 0 for i = np + nm + 1, . . . , n. Then, we have
P c¯ (x) = −
δi u(eki , x∗ + δ)
i
P i
.
x∗ k
δi xki
(A.5)
i
For the numerator of the right-hand side of Eq. (A.5), we have the following inequality: n X
np X
δi u(eki , x + δ) ≤ 2a¯ ∗
!2 δi
(A.6)
i =1
i=1
where a¯ = maxi,j aij . For the denominator of the right-hand side of Eq. (A.5), we have the following inequality: np n X X x∗k δi i < − (δi2 ). i=1
xki
(A.7)
i=1
Thus, we have
np 2 P δi c¯ (x) < 2a¯
i=1 np P
.
(A.8)
(δ ) 2 i
i=1
Moreover, since a¯ > 0 and
Pnp
i=1
n p 2 P δi 2a¯ np − 2a¯
i =1 np
P
(δi2 )
i =1
=
(δi2 ) > 0, we have
2a¯ np P
(δi2 )
np np X X np (δi2 ) − δi
i=1
δi )2 /
Pnp
i =1
!2
≥ 0.
(A.9)
i=1
Pnp
Thus, 2a¯ (n − 1) ≥ 2a¯ np ≥ 2a¯ (
i =1
i =1
(δi2 ) > c¯ (x) holds, and 2(n − 1) maxi,j (aij ) is an upper bound of c¯ (x).
A.3. Proof of Theorem 4 Let x∗ be a target state. Suppose that x∗i = 0, where x∗i is the ith element of x∗ . In this case, the dynamics of xi is given by x˙ i = {u(ei , x) − u(x, x) − c }xi . Therefore, if c > u(ei , x) for all x ∈ ∆, that is, if c > maxj aij , then xi (t ) with xi (0) ∈ (0, 1] decreases and converges to x∗i = 0 as t → ∞. A.4. Proof of Corollary 1 Theorem 4 shows that the target state is a global attractor. Moreover, the proof of Theorem 4 implies that xi decreases monotonically for all strategies i 6∈ C (x∗ ) and xj increases monotonically for the strategy i ∈ C (x∗ ). Therefore, the target state is Lyapunov stable since the ∞-norm of x − x∗ for any initial state x(0) ∈ ∆ decreases monotonically as time elapses. Thus, the target state is a globally asymptotically stable equilibrium point of Eq. (8) for any tax c > maxi∈S \{k} maxj∈S aij . References [1] R.M. Dawes, Social dilemmas, Annual Review of Psychology 31 (1980) 169–193. [2] C. Hauert, Spatial effects in social dilemmas, Journal of Theoretical Biology 240 (4) (2006) 627–636. [3] J. Tanimoto, H. Sagara, Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game, Biosystems 90 (1) (2007) 105–114. [4] H. Ohtsuki, M.A. Nowak, The replicator equation on graphs, Journal of Theoretical Biology 243 (1) (2006) 86–97. [5] J.M. Pacheco, A. Traulsena, M.A. Nowak, Active linking in evolutionary games, Journal of Theoretical Biology 243 (3) (2006) 437–443. [6] N.G. Mankiw, Principles of Economics, 4th ed., Thomson South-Western, Mason, 2007. [7] T. Kanazawa, T. Ushio, H. Goto, Replicator dynamics with government’s intervention by collection and reallocation of payoffs, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E90-A (10) (2007) 2170–2177. [8] J.W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, 1995. [9] T. Roughgarden, Selfish Routing and the Price of Anarchy, MIT Press, 2005. [10] T. Roughgarden, Routing games, in: N. Nisan, T. Roughgarden, É Tardos, V.V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, 2007, pp. 461–486. Ch. 18. [11] S Fischer, B Vöcking, On the evolution of selfish routing. In: Proceedings of the 12th European Symposium on Algorithms. 2004. p. 323–334.
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Replicator dynamics with Pigovian subsidy and capitation tax Takafumi Kanazawa, Yasuhiko Fukumoto, Toshimitsu Ushio ∗ , Takurou Misaka Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan
article Keywords: Evolutionary game Replicator dynamics Tax and subsidy
info
abstract The selfish behaviors of individuals can cause an inefficient society in which the total amount of the individuals’ utilities is not maximized. To resolve the problem, a government sometimes tries to control the population by imposing a tax on and/or offering a subsidy to the individuals who belong to the population. The tax is roughly classified into rate taxations and capitation taxes. Using an evolutionary game theory, the authors have proposed a differential equation model with rate taxations to analyze their effects on players’ behaviors. In this paper, we propose a differential equation model with the capitation taxes, and derive stability conditions of the target state. Moreover, we also discuss an application of our model to selfish routing games. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In real world, there exist many populations which consist of a large number of selfish players interacting with each other. In such a population, the purpose of each player often conflicts with the total purpose of the population, and a problem called a social dilemma occurs [1]. Evolutionary game theory has been used as a powerful mathematical framework to analyze social dilemma [2], and the dilemma game is characterized by replicator dynamics in Ref. [3]. Such a conflict is caused by several exogenous factors. Recently, a replicator equation on graphs [4] and a model of active linking [5] have been proposed to analyze effects of players’ interaction structures which is one of the exogenous factors in evolutionary games. In the replicator equation on a regular graph, each node represents a player and each edge means that players corresponding to its nodes can play a game. In the model of active linking, players form interaction links which have specific life-times at different rates and their dynamics are proposed. In these models, the interaction structures are modeled by transformations of the payoff matrices, and their effects on evolutions of cooperations are discussed [4,5]. To resolve the conflict between each player and the population, another exogenous factor called externality has to be taken into account. The word externality used in economics is an effect of a benefit obtained without paying payable costs or a cost paid without obtaining a receivable benefit [6]. Since the social dilemmas involve externalities, they cannot be resolved by the personal effort of each player. A government which has the comprehensive perspective is required for governing the population. In real world, it corresponds to the rulers such as governments of countries or cities, and executives of organizations or companies. To correct the externalities, a government sometimes tries to control the population by imposing a tax on and/or offering a subsidy to players who belong to the population. Such taxes and subsidies are called Pigovian taxes and Pigovian subsidies, respectively [6]. The tax is roughly classified into rate taxations and capitation taxes. In the former, the tax is determined based on payoffs the players earn while it is fixed in the latter. Using an evolutionary game theory, the authors have proposed a differential equation model with rate taxations to analyze the effects of rate taxations on players’ behaviors [7]. In this model, the government is willing to lead the population state to a desirable target state by using rate taxations and subsidies which
∗
Corresponding author. Tel.: +81 6 6850 6390; fax: +81 6 6850 6390. E-mail address: [email protected] (T. Ushio).
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depend on the target state. The amount of taxes which are imposed on each player is proportional to its payoff in the rate taxation model. Different from the rate taxation, the capitation taxes are determined independent of the players’ payoffs and the same amount of taxes is imposed on each player. In this paper, we deal with the capitation tax. To analyze effects of capitation taxes on players’ behaviors, we propose replicator dynamics with capitation taxes and subsidies. In this model, the government is willing to control the population state to a desirable target population state by collecting capitation taxes from all players and reallocating them as subsidies depending on the target state. We derive several properties of our model and provide the capitation tax which can make the target state locally or globally asymptotically stable. Moreover, we also discuss an application of our model to selfish routing games. 2. Preliminaries Consider a population which consists of a large number of players. Each player adopts a strategy which is inherited from its parent. In evolutionary games, repeatedly, two players are randomly drawn from the population to play a game with, and players produce offspring whose numbers depend on payoffs of the game [8]. In the context of social systems, this selection process is perceived as a model where players change their strategies according to the payoffs which depends on the results of the games. In this paper, we use the following notations: S = {1, . . . , n}: the set of players’ pure strategies; ei ∈ Rn (i = 1, . . . , n): a unit vector which corresponds to a pure strategy i; ∆ ⊂ Rn : the mixed strategy space (a convex combination of ei for all i ∈ S); int(∆): the interior of the mixed strategy space ∆; bd(∆): the boundary of the mixed strategy space ∆; C (x)(:= {i ∈ S |xi > 0}) ⊆ S: the carrier of the mixed strategy x; x = (x1 , . . . , xn )T ∈ ∆: a population state (a distribution of strategies, where xi is a proportion of players with strategy i); • A = (aij ) ∈ Rn×n : the payoff matrix; and • u(x, y) := xT Ay: the payoff function of players with mixed strategy x against y.
• • • • • • •
From the above definitions, conventional replicator dynamics is formulated as: x˙ i = {u(ei , x) − u(x, x)}xi
(1)
for each strategy i ∈ S. 3. Rate taxations We consider that the government collects a part of players’ payoffs at a taxation rate α ∈ [0, 1] and reallocating them as subsidies depending on the target state. If α = 0, then the government takes no taxes from the players. Such a situation is called non-intervention. On the other hand, if α = 1, then all payoffs which players earn are collected and reallocated by the government. Such a situation is called complete intervention. Suppose P that the number of players who adopt a strategy i ∈ S is pi > 0 and the number of all players in the population is p = i pi . If the government imposes the tax at rate of α on P each player, then the total payoff which is collected from players is α i pi u(ei , x). Denoted by x∗ ∈ ∆ is a target state which the government considers as a desirable population state. Suppose that the more desirable strategy for the government players adopt, the P more subsidies they are provided. Then, P thePsubsidy which is provided to players with a strategy i is assumed to be x∗i α j pj u(ej , x). Note that a summation i x∗i α j pj u(ej , x) of the reallocated payoffs to all players is equal P to the total tax which is collected from players since i x∗i = 1 holds. The reallocated subsidy to each strategy is equallydivided P to players with the strategy. P Therefore, the subsidy which is provided to each player with a strategy i is given by α x∗i j pj u(ej , x)/pi = α(x∗i /xi ) j∈S xj u(ej , x). Thus, a payoff function for the players with rate taxations and subsidies is given by the following function:
(1 − α)u(ei , x) + α
x∗i xi
u(x, x).
(2)
The first term of Eq. (2) is uncollected part of payoffs and the second one is collected and reallocated part of payoffs. Note that, since we assume that all collected taxes are reallocated in this case, the average payoff of players u(x, x) is independent of the taxation rate α . Moreover, suppose that every pure strategy is adopted by some players at least in the initial state, that is, xi (0) > 0 is assumed to hold for any pure strategy i ∈ S. In other words, we assume that x(0) ∈ int(∆). By this assumption, within any finite-time interval, xi > 0 holds for all i ∈ S and Eq. (2) is well defined. In this model, we assume that u(ei , x) ≥ 0 holds for any i ∈ S, and there exists j ∈ S satisfying u(ej , x) > 0 for all x ∈ ∆. Under this assumption, u(x, x) > 0 holds for any x ∈ ∆. Substituting Eq. (2) for players’ payoff function u(ei , x) of Eq. (1), we have replicator dynamics with rate taxations and subsidies as the following equation: for each strategy i ∈ S, x˙ i = (1 − α) xi {u(ei , x) − u(x, x)} + α x∗i − xi u(x, x),
(3)
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where α ∈ [0, 1] is the taxation rate. Taxation rates which can make the target state locally or globally asymptotically stable has been proposed in Ref. [7]. We summarize properties of the rate taxation model. By Eq. (3), obviously, if the government adopts the strategy non-intervention (α = 0), then Eq. (3) is reduced to Eq. (1). On the other hand, it has been proved if the government adopts the strategy complete intervention (α = 1), then the target state x∗ ∈ ∆ is a globally asymptotically stable equilibrium point. For the target state x∗ , we obtain the following proposition [7]: Proposition 1. If the target state x∗ ∈ ∆ is an equilibrium point of Eq. (1), then it is an equilibrium point of Eq. (3) for any α ∈ [0, 1]. On the other hand, if x∗ is not an equilibrium point of Eq. (1), then there does not exist α ∈ [0, 1) such that it is an equilibrium point of Eq. (3). Proposition 1 means if x∗ is not an equilibrium point of Eq. (1), then the government has to adopt complete intervention for leading the population state to the target state. Therefore, we consider the case that the target state is an equilibrium point of Eq. (1). We define a subset W ⊆ ∆, a scaled space Uε ⊆ ∆ of ∆ with the scaling factor ε ∈ (0, 1] and the center x∗ , and a function α( ˜ x) of x ∈ W as follows:
) X ∗ W = x ∈ ∆ x u(ei , x) < u(x, x) , i∈S i Uε = x¯ ∈ ∆ x¯ = (1 − ε)x∗ + ε x, x ∈ ∆ , P ∗ u(x, x) − xi u(ei , x) i∈S o . α( ˜ x) = P n x∗ ∗ xi xi u(x, x) − u(ei , x) i (
(4) (5)
(6)
i∈S
The target state x∗ is a locally asymptotically stable equilibrium point of Eq. (3) for all α ∈ (α, 1], where α = 0 if there exists ε > 0 satisfying Uε ∩ W = ∅, otherwise α = supx∈Uε ∩W α( ˜ x) for some ε > 0 satisfying Uε ∩ W 6= ∅. Since Uε equals to ∆ with ε = 1, we obtain the following theorem [7]: Theorem 1. We set α = 0 if W = ∅ and α = supx∈W α( ˜ x) if W 6= ∅. Then, x∗ is a globally asymptotically stable equilibrium point of Eq. (3) for all α ∈ (α, 1]. 4. Capitation taxes 4.1. Model In this section, we deal with the capitation taxes. Different from the rate taxation, the capitation taxes are determined independent of the players’ payoffs. To analyze their effects on players’ behaviors, we propose replicator dynamics with capitation taxes and subsidies. Suppose P that the number of players who adopt a strategy i ∈ S is pi > 0 and the number of all players in the population is p = i pi . If the government imposes the capitation tax c on each player, then the total payoff which is collected from players is cp. Let x∗ be a target state. Suppose that the more desirable strategy for the government players adopt, the more subsidies they are provided. Then, the subsidy which is provided to players with a strategy i is assumed to be cpx∗i . The reallocated subsidy to each strategy is equally-divided to players with the strategy. Therefore, the subsidy which is provided to each player with a strategy i is given by cpx∗i /pi = cx∗i /xi . Thus, the payoff function for the players with subsidies and capitation taxes is given by the following function: u(ei , x) − c + c
x∗i xi
= u(ei , x) − c 1 −
x∗i xi
.
(7)
Similar to the model of the rate taxations, we assume that all collected taxes are reallocated and the average payoff of players is independent of the capitation tax c. We assume that x(0) ∈ int(∆) and Eq. (7) is well defined within any finitetime interval. Substituting the right-hand side of Eq. (7) for players’ payoff function u(ei , x) of Eq. (1), we have replicator dynamics with subsidies and capitation taxes as follows: for each strategy i ∈ S, x˙ i = {u(ei , x) − u(x, x)}xi + c (x∗i − xi ),
(8)
where c ≥ 0 is the amount of the capitation taxes. Note that we allow the case that the tax c is greater than the players’ original payoffs, that is, we allow c > u(ei , x). Eq. (8) is given by adding the negative feedback term c (x∗i − xi ) to the conventional replicator dynamics equation (1), and the tax c is considered as a feedback gain.
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4.2. Properties We show several properties of Eq. (8) and prove a condition for the capitation tax to make the target state locally or globally asymptotically stable. See Appendix for proofs of propositions, theorems, and a corollary in this section. Proposition 2 (Invariance Under a Local Shift). Eq. (8) is invariant under a local shift of the payoff matrix A, where the local shift is the addition of a constant to all elements of a column of A. Eq. (1) is invariant under the local shift of the payoff matrix A, but the rate taxation model proposed by [7] is changed by the local shift. Using the invariance under the local shift, without loss of generality, we can suppose that each element of the payoff matrix A is non-negative. Especially, in two-strategy game, we can simplify the matrix A by setting all non-diagonal elements to zero. Proposition 3 (Equilibrium Target Point). If the target state x∗ is an equilibrium point of Eq. (1), then it is an equilibrium point of Eq. (8) for any tax c > 0. Generally, it is not always true that the target state x∗ is an equilibrium point of Eq. (8). However, if the target state x∗ is an equilibrium point of Eq. (1), then x∗ is always an equilibrium point of Eq. (8). In this paper, we focus on the case that the target state x∗ is an equilibrium point of Eq. (1), and discuss its stability. Theorem 2 (Locally Asymptotic Stability). Let the linearization system of Eq. (1) at the target state x = x∗ be x˙ = J0 x, and the eigenvalues of the Jacobian matrix J0 of Eq. (1) at x = x∗ be λ0i (i = 1, . . . , n). Then, the linearization system of Eq. (8) at the target state x = x∗ is given by x˙ = (J0 − cIn )x,
(9)
where In is the n × n unit matrix. The origin is asymptotically stable in Eq. (9) if and only if c > maxi {R(λ0i )}, where R(λ0i ) is the real part of λ0i . If the origin of the linearization system equation (9) is asymptotically stable, then the target state x = x∗ of the nonlinear system equation (8) is locally asymptotically stable. In the case that c = maxi {R(λ0i )} holds, although the origin of Eq. (9) is Lyapunov stable, the stability of x = x∗ depends on the higher-order terms of Eq. (8). Theorem 3 (Globally Asymptotic Stability). Suppose that each element of the payoff matrix A is non-negative and define c¯ (x) as follows: for all x ∈ int(∆) \ {x∗ }, n P
c¯ (x) =
(xi − x∗i )u(ei , x)
i=1 n P i=1
x∗
.
(10)
(x∗i − xi ) xii
If the target state x∗ is a Nash equilibrium, then it is a globally asymptotically stable equilibrium point of (8) for c > max{0, supx∈int(∆)\{x∗ } c¯ (x)}. Moreover, c¯ (x) < 2(n − 1) maxi,j aij for any x ∈ int(∆) \ {x∗ }. Due to Proposition 2, for a game with a payoff matrix A˜ which has a negative element, we transform it to a payoff matrix A whose elements are non-negative by the local shift to the matrix A˜ and apply Theorem 3 to the matrix A. The target state of the game with A˜ is also stabilized globally by the tax c for the game with A. Now, we intervene a population state x as a situation that all players of the population adopt a mixed strategy x. Then, the numerator of c¯ (x) is a difference between a payoff of a player with a mixed strategy x and that of a player with x∗ . c¯ (x) is positive if the former is larger than the latter while it is negative if the latter is larger than the former. Since players have no incentive to change their strategy x to x∗ if the former is larger, the achievement of the target state x∗ requires the government’s intervention. The larger the difference is, the larger the tax c which can stabilize the target state x∗ is. On the other hand, if the latter is larger, then players are willing to change their strategy x to x∗ independent of the tax c. The denominator of c¯ (x) estimates a kind of distance between a population state x and the target state x∗ . Since the distance is not a simple summation of differences between xi and x∗i but a weighted summation of the differences multiplied by x∗i /xi , the positive differences are amplified and the negative differences are discounted. So, roughly speaking, the farther a population state x is from the target state x∗ , the larger the denominator is. To set c = 2(n − 1) maxi,j aij as a tax means that the government imposes the larger tax on players when the number of strategies and the maximum payoff are larger. Theorem 3 supposes that the target state x∗ is a Nash equilibrium. The target state which is not a Nash equilibrium must be on the boundary of ∆, that is, x∗ ∈ bd(∆). For such a target state, we have the following theorem: Theorem 4. Suppose that each element of the payoff matrix A is non-negative and x∗ ∈ bd(∆). For all i 6∈ C (x∗ ) and for any c > maxi6∈C (x∗ ) maxj∈S aij , orbits x(t ) of Eq. (8) satisfy limt →∞ xi (t ) = 0.
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Fig. 1. Phase portrait for c = 0.
When x∗i = 0 and xi (0) = 0, the set {x ∈ ∆|xi = 0} ⊂ bd(∆) is a positive invariant set since x˙ i (t ) ≡ 0. Therefore, it is expected that we can find the tax c which makes the target state an attractor of Eq. (8) by the following procedure. First, using Theorem 4, find a tax c1 which converges xi to zero for all strategies i 6∈ C (x∗ ). Next, for a sub-game which omits strategies i 6∈ C (x∗ ), find a tax c2 which stabilizes the target state x∗ globally using Theorem 3. Finally, we set c = max{c1 , c2 } as a tax so that the target state x∗ becomes an attractor of Eq. (8) whose basin is ∆. However, it is not obvious that the tax c which is given by the above procedure makes the target state x∗ ∈ bd(∆) an attractor of Eq. (8) since the orbits of Eq. (8) cannot converge to the set {x ∈ ∆|xi = 0} ⊂ bd(∆) within finite-time intervals. Even if the tax can make the target state an attractor, the target state may not be Lyapunov stable. When the target state x∗ is a vertex of ∆, we can prove x∗ is not only an attractor but also a Lyapunov stable equilibrium point, that is, x∗ is a globally asymptotically stable equilibrium point. Corollary 1 (Stability of the Target State on a Vertex). Suppose that the target state x∗ is a vertex of ∆. Then it is a globally asymptotically stable equilibrium point of Eq. (8) if c > maxi∈S \{k} maxj∈S aij . 4.3. Example Heavy traffic in urban areas cause several kinds of problems such as air pollution, increase in CO2 level, and so on. Recently, park and ride (P&R) is paid much attention to as a useful system for a traffic congestion problem. Park and ride means that people who wants to go the urban areas drive their cars by a train station or a bus stop, park their cars there, and transfer to a public transport system such as a train or a bus. As the other option to reduce traffic congestion, they use bicycles. In this section, suppose that the strategy 1, 2, and 3 correspond to go by ‘‘Car’’, ‘‘P&R’’, and ‘‘Bicycle’’, respectively. We set players’ payoff matrix A as follows: 1 7 6
" A=
8 2 5
9 4 . 3
#
(11)
Fig. 1 shows its phase portrait. There exists a unique stable equilibrium point ◦, three unstable equilibrium points •, and three saddle points in Fig. 1. In the case that the government does not impose taxes, all orbits whose initial states are in int(∆) converge to the unique interior stable equilibrium point. Suppose that the government perceives the case that no one uses his/her car is the desirable population state, that is, we consider the saddle point (0, 1/4, 3/4) as the target state x∗ . Consider the linearization system at the target state x∗ . Note that x always satisfies x1 + x2 + x3 = 1. We can eliminate x3 by substituting 1 − x1 − x2 for x3 . Since the eigenvalues of the Jacobian matrix at x∗ are 21/4 and −3/4, Theorem 2 shows that the target state is a locally asymptotically stable equilibrium point if c > 21/4. Figs. 2 and 3 show transient controlled behaviors with c = 26/5 (<21/4) and c = 53/10 (>21/4), respectively. In the case of c = 26/5, all orbits converge to a neighborhood of the target state, but do not converge to the target state itself as shown in Fig. 2. However, in the case of c = 53/10, all orbits converge to the target state as shown in Fig. 3. Theorem 4 shows that all orbits converge to a surface with x1 = 0 where the target state x∗ ∈ bd(∆) is if c > 9. A two-strategy sub-game which omits the strategy 1 has a unique locally asymptotically stable equilibrium point (x2 , x3 ) = (1/4, 3/4) if c = 0. Theorem 3 shows that the target state is globally stabilized if c > 0. Depicted in Fig. 4 is a phase portrait for c = 91/10, which shows that the target state is globally asymptotically stable. 5. An application of capitation tax model to selfish routing A selfish routing game is a simple model of selfish behaviors in networks [9,10]. Its replicator dynamics has been proposed [11]. In such a game, there exists inefficiency caused by selfish route selections of each player and several
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Fig. 2. Transient behavior of x1 for c = 26/5.
Fig. 3. Transient behavior of x1 for c = 53/10.
Fig. 4. Phase portrait for c = 91/10.
methodologies have been proposed to reduce it. In this section, we consider an application of the capitation tax model to such a selfish routing game. In selfish routing games, suppose that there is a fixed flow demand and it is routed from a source to a sink. A player’s strategy is a selection from all possible paths from the source to the sink, and its payoff is related to latency on each path. Since the player selects a path in order to minimize the latency, the payoff of selecting a path i ∈ S is set to u(ei , x) = −li (x), where li (x) is latency of a path i with flow x ∈ ∆. We can consider the capitation tax c in our proposed model is additional fictitious delay to flow of each path in routing games. On the other hand, we cannot introduce the subsidies into the selfish routing since we cannot reduce latency in network by the control. Therefore, using Proposition 2, we modify our proposed payoff function with subsidies and capitation taxes given by Eq. (7) as follows:
u(ei , x) − max c i∈S
x∗i xi
+c
x∗i xi
.
(12)
Substituting Eq. (12) for player’s payoff function u(ei , x) of Eq. (1), we have Eq. (7). Since Eq. (12) is always less than or equal to u(ei , x) for all i ∈ S with equality if and only if x = x∗ , it is well defined as players’ payoff functions in selfish routing games.
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Fig. 5. Braess’ Paradox. Each edge is labeled with its latency, where xei is the amount of traffic using the edge which is labeled with it, that is, xe1 = x1 + x2 and xe2 = x2 + x3 .
Fig. 6. Phase portrait for c = 0.
As a simple example, we consider a network shown in Fig. 5. Suppose that there is a fixed flow demand 1 and it is routed from source s to sink t. This network has three s–t paths. Let strategies 1, 2, and 3 correspond to a selection of the paths s–v –t, s–v –w –t, and s–w –t, respectively. The payoff matrix is given by latency of each edge of the network shown in Fig. 5. By a local shift of the given payoff matrix, we have the following payoff matrix whose elements are non-negative: 0 1 1
" A=
0 0 0
1 1 . 0
#
(13)
In this network, a Nash equilibrium flow is x = (0, 1, 0), while the minimum latency flow is (1/2, 0, 1/2). Such a situation is well known as Braess’s Paradox. We consider the minimum latency flow (1/2, 0, 1/2) as the target state. Consider the linearization system at the target state x∗ . The eigenvalues of the Jacobian matrix at x∗ are 1/2 and −1/2. Theorem 2 shows that the target state is a locally asymptotically stable equilibrium point if c > 1/2. Moreover, Theorem 4 shows that all orbits converge to a surface with x2 = 0 if c > 1. A two-strategy sub-game which omits the strategy 2 has a unique locally asymptotically stable equilibrium point (x1 , x3 ) = (1/2, 1/2) for any c ≥ 0, and Theorem 3 shows that the target state is globally stabilized if c > 0. Figs. 6 and 7 show phase portraits of the selfish routing with taxes and subsidies for c = 0 and c = 11/20 (>1/2), respectively. In this example, by Theorem 2, a tax satisfying the condition c > 1/2 stabilizes the target state globally as shown in Fig. 7. By Theorem 4, all orbits converge to a surface with x2 = 0 if a tax satisfies the condition c > 1. 6. Conclusions In this paper, we have proposed replicator dynamics with capitation taxes and subsidies to analyze effects of capitation taxes on players’ behaviors. In this model, the government’s purpose is to control the population state to a target state by collecting capitation taxes and reallocating them as subsidies depending on the target state. We have proved several properties of our model and have provided a capitation tax which can make the target state locally or globally asymptotically stable. Moreover, we have also discussed an application of our model to selfish routing games. In our model, the amount of taxes is assumed to be independent of the population state. However, we can consider that it is changed depending on the population state as a government’s policy. It is our future work to propose a model which can deal with such changes of capitation taxes.
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e825
Fig. 7. Phase portrait for c = 11/20.
Acknowledgments This research was supported in part by KAKENHI (No. 19860045) and ‘‘Global COE (Centers of Excellence) Program’’ of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Appendix. Proofs A.1. Proof of Theorem 2 Let the linearization system of Eqs. (1) and (8) at the target state x = x∗ be x˙ = J1 (c )x and x˙ = J0 x, respectively. Moreover, we denote Eqs. (1) and (8) by x˙ i = fi (x) and x˙ i = fi0 (x), respectively. Obviously, x˙ i = fi0 (x) = fi (x) + c (x∗i − xi ) holds. Then, we have
∂f 0 1
···
∂ x1 . J1 (c ) = .. ∂f 0 n
..
.
···
∂ x1
∂f
∂ f10 ∂ xn .. . 0 ∂f
1
∂ x1 .. = . ∂f
n
∂ xn
−c
··· ..
n
···
∂ x1
x=x∗
.
∂ f1 ∂ xn .. .
∂ fn −c ∂ xn
,
(A.1)
x =x ∗
that is, J1 (c ) = J0 − cIn . Therefore, if we suppose that the eigenvalues of J1 (c ) are λ11 , . . . , λ1n , then λ1i = λ0i − c holds for i = 1, . . . , n. Note that λ0i is a constant which depends on the payoff matrix A and the target state x∗ . Thus, the real parts of all eigenvalues of J1 (c ) are negative, that is, the origin of Eq. (9) is an asymptotically stable equilibrium point if and only if c > maxi {R(λ0i )} holds. A.2. Proof of Theorem 3 Consider the following function: V (x) =
X
∗
−xi log
i x∗ >0 i
xi x∗i
.
(A.2)
This function satisfies V (x) ≥ 0 with equality if and only if x = x∗ . The time derivative of V (x) along solutions of Eq. (8) is V˙ (x) = −
X
(x∗i − xi )u(ei , x) −
i
X x∗ (x∗i − xi ) i c . i
xi
(A.3)
Then V˙ (x) < 0 for all x 6= x∗ if
( c > max 0,
) sup
x∈int(∆)\{x∗ }
(¯c (x)) .
Thus, x∗ is a globally asymptotically stable equilibrium point.
(A.4)
e826
T. Kanazawa et al. / Nonlinear Analysis 71 (2009) e818–e826
We will prove that 2(n − 1) maxi,j (aij ) is an upper bound of c¯ (x). Let δ = (δ1 , . . . , δn ) be a difference between x and x∗ , that is, δi = xki − x∗ki for i = 1, . . . , n. Without loss of generality, we assume that δi > 0 for i = 1, . . . , np , δi < 0 for i = np + 1, . . . , np + nm , and δi = 0 for i = np + nm + 1, . . . , n. Then, we have
P c¯ (x) = −
δi u(eki , x∗ + δ)
i
P i
.
x∗ k
δi xki
(A.5)
i
For the numerator of the right-hand side of Eq. (A.5), we have the following inequality: n X
np X
δi u(eki , x + δ) ≤ 2a¯ ∗
!2 δi
(A.6)
i =1
i=1
where a¯ = maxi,j aij . For the denominator of the right-hand side of Eq. (A.5), we have the following inequality: np n X X x∗k δi i < − (δi2 ). i=1
xki
(A.7)
i=1
Thus, we have
np 2 P δi c¯ (x) < 2a¯
i=1 np P
.
(A.8)
(δ ) 2 i
i=1
Moreover, since a¯ > 0 and
Pnp
i=1
n p 2 P δi 2a¯ np − 2a¯
i =1 np
P
(δi2 )
i =1
=
(δi2 ) > 0, we have
2a¯ np P
(δi2 )
np np X X np (δi2 ) − δi
i=1
δi )2 /
Pnp
i =1
!2
≥ 0.
(A.9)
i=1
Pnp
Thus, 2a¯ (n − 1) ≥ 2a¯ np ≥ 2a¯ (
i =1
i =1
(δi2 ) > c¯ (x) holds, and 2(n − 1) maxi,j (aij ) is an upper bound of c¯ (x).
A.3. Proof of Theorem 4 Let x∗ be a target state. Suppose that x∗i = 0, where x∗i is the ith element of x∗ . In this case, the dynamics of xi is given by x˙ i = {u(ei , x) − u(x, x) − c }xi . Therefore, if c > u(ei , x) for all x ∈ ∆, that is, if c > maxj aij , then xi (t ) with xi (0) ∈ (0, 1] decreases and converges to x∗i = 0 as t → ∞. A.4. Proof of Corollary 1 Theorem 4 shows that the target state is a global attractor. Moreover, the proof of Theorem 4 implies that xi decreases monotonically for all strategies i 6∈ C (x∗ ) and xj increases monotonically for the strategy i ∈ C (x∗ ). Therefore, the target state is Lyapunov stable since the ∞-norm of x − x∗ for any initial state x(0) ∈ ∆ decreases monotonically as time elapses. Thus, the target state is a globally asymptotically stable equilibrium point of Eq. (8) for any tax c > maxi∈S \{k} maxj∈S aij . References [1] R.M. Dawes, Social dilemmas, Annual Review of Psychology 31 (1980) 169–193. [2] C. Hauert, Spatial effects in social dilemmas, Journal of Theoretical Biology 240 (4) (2006) 627–636. [3] J. Tanimoto, H. Sagara, Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game, Biosystems 90 (1) (2007) 105–114. [4] H. Ohtsuki, M.A. Nowak, The replicator equation on graphs, Journal of Theoretical Biology 243 (1) (2006) 86–97. [5] J.M. Pacheco, A. Traulsena, M.A. Nowak, Active linking in evolutionary games, Journal of Theoretical Biology 243 (3) (2006) 437–443. [6] N.G. Mankiw, Principles of Economics, 4th ed., Thomson South-Western, Mason, 2007. [7] T. Kanazawa, T. Ushio, H. Goto, Replicator dynamics with government’s intervention by collection and reallocation of payoffs, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E90-A (10) (2007) 2170–2177. [8] J.W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, 1995. [9] T. Roughgarden, Selfish Routing and the Price of Anarchy, MIT Press, 2005. [10] T. Roughgarden, Routing games, in: N. Nisan, T. Roughgarden, É Tardos, V.V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, 2007, pp. 461–486. Ch. 18. [11] S Fischer, B Vöcking, On the evolution of selfish routing. In: Proceedings of the 12th European Symposium on Algorithms. 2004. p. 323–334.