## Ke Xu , Ke Wu and Rongpei Zhou## |

Notations | Definitions |
---|---|

V | The set of all players (vertices) |

X | The finite strategy space |

[TeX:] $$X_i$$ | The strategy set of player [TeX:] $$i$$ |

μ | The penalty coefficient of uncovered edge |

[TeX:] $$\omega(\cdot)$$ | The probability weight function |

a | The rational coefficient |

[TeX:] $$\Gamma_i$$ | The neighbor set of player [TeX:] $$i$$ |

T | The superscript for transpose |

EUT | The expected utility theory |

PT | The prospect theory |

[TeX:] $$p_{i, 1}$$ | The probability of choosing strategy 1 for [TeX:] $$i$$ |

[TeX:] $$p_{i, 0}$$ | The probability of choosing strategy 0 for [TeX:] $$i$$ |

[TeX:] $$k_i$$ | The degree of player [TeX:] $$i$$ |

< k > | The average degree of network |

i) We present the player’ utility (cost) function based on prospect theory, and describe the vertex cover problem as a covering game model. Then, we further present a game-based distributed algorithm, i.e, the fictitious play algorithm, which can guarantee to obtain a Nash equilibrium, and it satisfies the vertex cover state of a network.

ii) The simulation results illustrate that deviation from objectivity can strongly impact the overall cover level of networks. Specifically, the more serious the deviation from objectivity, the worse the cover level, and the slower the convergence rate of the fictitious play algorithm.

The paper is organized as follows. In Section II, we introduce prospect theory and formulate a covering game model. In Section III, we explore the relationship between the vertex cover and Nash equilibrium of the formulated game model. In Section IV, we present a fictitious play distributed algorithm, and prove that the states of all vertices converge to a Nash equilibrium. In Section V, numerical simulation results are presented to demonstrate the effectiveness of the presented algorithm, and show the impact of subjectivity on cover results. Finally, conclusions of this paper are drawn in Section VI. Note that the notations used in this paper are listed in Table I.

A. Prospect Theory

Expected utility theory (EUT) allows each player to evaluate an objective expected utility, and then makes corresponding decision. However, in the real-life scenarios, players’ behavior will deviate from the rational behavior predicted by the EUT. This phenomenon is attributed to the risk and uncertainty that players often face when making decisions.

Fortunately, the prospect theory (PT) can be used to model how players behave irrationally under the risk and uncertainty circumstances [12]. An important concept of the PT is proba- bility weight effect, which converts objective probability into subjective probability, so as to establish a prospect game model of players. For probability weight effect, it reveals that each player [TeX:] $$i$$ adopts his subjective probability [TeX:] $$w_i\left(p_i\right)$$ rather than objective probability [TeX:] $$p_i$$, where subjective probability [TeX:] $$w_i\left(p_i\right)$$ is a nonlinear transformation from [TeX:] $$p_i$$ to [TeX:] $$w_i\left(p_i\right)$$ [23]–[25]. Without loss of generality, we can assume that [TeX:] $$w_i\left(p_i\right)$$) is the same for all players, i.e., [TeX:] $$w_i\left(p_i\right)$$ = [TeX:] $$w\left(p_i\right)$$. At present, the Prelec function is widely used as a classical weight function [26], which is described as

where [TeX:] $$\sigma$$ and [TeX:] $$\omega(\sigma)$$ are the objective and subjective probability, respectively; α ∈ (0,1] is a rational coefficient, which reveals how a player subjective evaluation distorts the objective probability. The greater the rational coefficient α, the higher the rational degree. It is worth noting that when α = 1, the probability weight effect will coincide with the probability of the EUT. An illustration on the impact of α is shown in Fig. 1.

From Fig. 1, we can observe that the probability weight function [TeX:] $$\omega(\sigma)$$ has some properties, include: i) It is asymmetric with fixed point and inflection point at (1/e,1/e); ii) When 0 < σ < 1/e, ω(σ) > σ, moreover, 1/e < σ < 1, ω(σ) < σ; iii) ω(0) = 0, ω(1) = 1; iv) It is concave if 0 < σ ≤ 1/e and convex if 1/e < σ ≤ 1; v) A small α results in a more curved probability weight function.

B. Game Model

An undirected graph is defined by [TeX:] $$\Xi$$ = {V,E}, where V = {1,···,n} is the set of vertices and E = {[TeX:] $$e_{i j}$$} is the set of edges. For an undirected graph [TeX:] $$\Xi$$, [TeX:] $$e_{i j}$$ = 1 if an edge exists between vertex i and vertex j; otherwise, [TeX:] $$e_{i j}$$ = 0. A subset VVC of V is called a vertex cover if, for each edge [TeX:] $$e_{i j}\left(e_{i j} \in E\right)$$ has at least one endpoint (vertex) belongs to [TeX:] $$V_{\mathrm{VC}}$$. A minimum vertex cover [TeX:] $$V_{\mathrm{MVC}}$$ is a vertex cover [TeX:] $$V_{\mathrm{VC}}$$ with the minimum cardinality.

The vertex cover problem can be formulated as a covering game, denoted by [TeX:] $$G=\left\{V,\left\{X_i\right\}_{i \in V},\left\{u_i\right\}_{i \in V}, E\right\}$$ , where [TeX:] $$X_i$$ = {0, 1} (0 and 1 are the uncovered and covered strategies, respectively) is the strategy set of player (vertex) [TeX:] $$i, \forall i \in V$$ strategy profile of all players is [TeX:] $$x=\left(x_1, \cdots, x_n\right) \in X$, $x_i \in X_i$$; and [TeX:] $$u_i$$ is the utility (cost) function of player [TeX:] $$i$$ [5].

Based on the above definition, the cost function for a player [TeX:] $$i(i \in V)$$ is given by [27], [28] where [TeX:] $$x_{-i}=\left(x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_n\right)$$ is the strategy profile of all players except player [TeX:] $$i$$; The first item on the right of (2) represents the cost that player [TeX:] $$i$$ pays if he chooses the covered strategy. μ > 0 is penalty coefficient, the corresponding second item represents the penalty of an uncovered edge.

In the covering game G, we will model the player’s cost function based on the continuous strategies, rather than discrete and deterministic strategies. Intuitively, continuous strategy [TeX:] $$p_{i, x_i} \in[0,1]$$ in this paper is a probabilistic choice, which captures the frequency of players choosing the strategy [TeX:] $$x_i$$. In this case, let [TeX:] $$p=\left(p_{1,1}, \cdots, p_{n, 1}\right)$$ be the continuous strategy profile of all players, which is a row vector.

In the EUT, each player can objectively evaluate each neigh- bor’s probabilistic choice, and objectively select his continuous strategy [TeX:] $$p_{i, x_i} \in[0,1]$$ to optimize his cost. Thus, the cost of player [TeX:] $$i$$ under the EUT is given by

where [TeX:] $$p_{i, 1}$$ is the probability of choosing covered strategy 1 for player [TeX:] $$i ; p_{-i, 1}=\left[p_{1,1}, \cdots, p_{i-1,1}, p_{i+1,1}, \cdots, p_{n, 1}\right]$$ is the continuous strategy profile of all players other than player [TeX:] $$i$$.

In this paper, we consider applying the probability weight effect of prospect theory for modeling, that is, each player will subjectively evaluate the neighbor’s continuous strategy. Thus, the cost of player [TeX:] $$i$$ under the PT is given by

where [TeX:] $$\omega_i(\cdot)$$ is the Prelec function, which is shown in (1).

For each player [TeX:] $$i$$, he plays the covering game by solving the following optimization problem as

where [TeX:] $$U_i(\cdot)$$ can be either the EUT cost (3) or the PT cost (4).

Definition 1 : A Nash equilibrium [TeX:] $$p^*=\left(p_{1,1}^*, \cdots, p_{n, 1}^*\right)$$ is such the continuous strategy profile where no player can decrease his cost by changing his continuous strategy [29], i.e.,

where [TeX:] $$\forall p_{i, 1}^{\prime} \neq p_{i, 1}^*$$. Note that the Nash equilibrium (NE) defined in (6) is applicable for both the EUT and the PT, the difference would be in whether one is using (3) or (4), respectively.

Note that for any continuous strategic game, there exists at least one NE under both the EUT and the PT. Define [TeX:] $$p^{\mathrm{EUT}}=\left(p_{1,1}^{\mathrm{EUT}}, \cdots, p_{n, 1}^{\mathrm{EUT}}\right) \text { and } p^{\mathrm{PT}}=\left(p_{1,1}^{\mathrm{PT}}, \cdots, p_{n, 1}^{\mathrm{PT}}\right)$$ be a 1,1 n,1 1,1 n,1 NE of the EUT and the PT, respectively. Next, I give an example to illustrate it.

Example 1: Consider three players to perform covering game, penalty coefficient μ is set to 1.6, adjacency matrix is

[TeX:] $$\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right] \text {, }$$

it is easy to check that there exists seven NEs regardless of the [TeX:] $$\alpha$$. For instance, if rational coefficient [TeX:] $$\alpha$$ = 1, and seven NEs are [TeX:] $$p^{\mathrm{EUT}, 1}$$ = (0, 1, 1), [TeX:] $$p^{\mathrm{EUT}, 2}$$ = (1, 0, 1), [TeX:] $$p^{\mathrm{EUT}, 3}$$ = (1, 1, 0), [TeX:] $$p^{\mathrm{EUT}, 4}$$ = (1, 0.375, 0.375), [TeX:] $$p^{\mathrm{EUT}, 5}$$ = (0.375,1,0.375), [TeX:] $$p^{\mathrm{EUT}, 6}$$ = (0.375,0.375,1), and [TeX:] $$p^{\mathrm{EUT}, 7}$$ = (0.6875, 0.6875, 0.6875).

Proposition 1: Consider a covering game G, if penalty coefficient μ > e/(e − 1), any NE [TeX:] $$p^*=\left(p_{i, 1}^*, \cdots, p_{n, 1}^*\right)$$ under both the EUT and the PT satisfies: if [TeX:] $$p_{i, 1}^* \leq 1 / e$$, then [TeX:] $$\forall j \in \Gamma_i, p_{i 1}^*>1 / e$$. Note that the NE [TeX:] $$p^*$$ can be either the EUT or the PT.

Proof: Suppose that [TeX:] $$\exists j^{\prime} \in \Gamma_i$$, such that [TeX:] $$p_{j^{\prime}, 1}^* \leq 1 / e$$. Let [TeX:] $$p_{i, 1}^{\prime}>1 / e$$.

Under the EUT, according to Definition 1, the following inequation

holds

Since [TeX:] $$p_{i, 1}^*-p_{i, 1}^{\prime}$$ < 0, and

then

[TeX:] $$U_i^{\mathrm{EUT}}\left(p_{i, 1}^*, p_{-i, 1}^*\right)-U_i^{\mathrm{EUT}}\left(p_{i, 1}^{\prime}, p_{-i, 1}^*\right)>0$$

which is a contradiction to (7). Thus, the hypothesis is not hold.

Under the PT, according to Definition 1, the following inequation

holds.

Since

[TeX:] $$1-\mu \sum_{j=1}^n\left(1-\omega_i\left(p_{j, 1}^*\right)\right) e_{i j} \leq 1-\mu(1-1 / e)<1-1=0$$

then

[TeX:] $$U_i^{\mathrm{PT}}\left(p_{i, 1}^*, p_{-i, 1}^*\right)-U_i^{\mathrm{PT}}\left(p_{i, 1}^{\prime}, p_{-i, 1}^*\right)>0$$

which is a contradiction to (8). Thus, the hypothesis is not hold. Hence, the proof is completed.

Further, we define a rounding function

if [TeX:] $$s_i\left(p_{i, 1}\right)$$ = 1, we consider that the state of player [TeX:] $$i$$ is a i i,1 covered state; otherwise, it is an uncovered state. Besides, define s = [TeX:] $$\left(s_1, \cdots, s_n\right)$$ be the state profile for all players, where [TeX:] $$s_i=s_i\left(p_{i, 1}\right)$$.

According to (9), a theorem is given as follows

Theorem 1: Consider a covering game G, if penalty coeffi- cient μ > e/(e − 1), any NE [TeX:] $$p^*$$ satisfies: if [TeX:] $$s_i\left(p_{i, 1}^*\right)$$ = 0, then [TeX:] $$\forall j \in \Gamma_i, s_j\left(p_{j, 1}^*\right)$$ = 1, where [TeX:] $$\Gamma_i$$ is the neighbor set of [TeX:] $$i$$.

Remark 1: The Theorem 1 shows that there no 0-0 edge (uncover edge) under certain constraint of penalty coefficient μ. In this case, the vertices’ state under any NE under Nash equilibrium satisfies vertex cover state of a general network. This relationship is only determined by the game model other than any network topology or any algorithm.

A. Fictitious Play Algorithm

In this subsection, we present a fictitious play distributed algorithm, which requires the minimum memory length 1, that is, each player only need his previous strategy (the probability t of choosing strategy 1. The algorithm is raised from the fictitious play (FP) [30]. Although there exist many distributed algorithms can find a NE [31]–[33], the FP distributed algo- rithm is well-known as a popular algorithm for players, which is a simple and practical algorithm that can keep an acceptable running time [34]. We present the FP distributed algorithm as follows and the pseudo-code in Table II.

At each time step [TeX:] $$t$$, the probability [TeX:] $$p_{i, 1}^t$$ of player [TeX:] $$i$$ is updated as

where [TeX:] $$p_{i, 1}^t$$ is the probability of player [TeX:] $$i$$ choosing strategy 1 at time step [TeX:] $$t$$, and [TeX:] $$v_{i, 1}^t$$ is determined by

where the [TeX:] $$U_i\left(x_i, p_{-i, 1}^{t-1}\right)$$ is the cost obtained by player [TeX:] $$i$$ with re- spect to the continuous strategy profile of the neighbors, when player [TeX:] $$i$$ chooses strategy [TeX:] $$x_i$$ at time step [TeX:] $$t$$. Besides, the cost [TeX:] $$U_i\left(x_i, p_{-i, 1}\right)$$ can be either cost of the EUT [TeX:] $$U_i^{\mathrm{EUT}}\left(x_i, p_{-i, 1}^{t-1}\right)$$ or cost of the PT [TeX:] $$U_i^{\mathrm{PT}}\left(x_i, p_{-i, 1}^{t-1}\right)$$.

The stopping criterion of the presented algorithm can be defined as

where [TeX:] $$Q$$ is an arbitrary large number (typically goes to tttT infinity), [TeX:] $$P_i^t=\left[p_{i, 1}^t, p_{i, 0}^t\right]^T$$.

B. Convergence Analysis

The presented algorithm can guarantee that any continuous strategy profile converges to an equilibrium state after a certain time steps [35]. Given any continuous strategy profile, suppose

Table II.

Algorithm 1: Fictitious play algorithm |
---|

Input: Undirected network [TeX:] $$\Xi$$; penalty coefficient [TeX:] $$\mu$$; rational coefficient [TeX:] $$a$$; initial continuous strategy profile [TeX:] $$p^0$$. Output: NE [TeX:] $$p^*$$; state profile [TeX:] $$s^*$$. repeat: Each player [TeX:] $$i$$ obtains neighbors’ continuous strategy profile\ at time [TeX:] $$t$$: [TeX:] $$p_{-i, 1}^{t-1}$$Calculate cost of each discrete strategy by (3), (4): [TeX:] $$U_i^{\mathrm{EUT}}\left(x_i, p_{-i, 1}^{t-1}\right), U_i^{\mathrm{PT}}\left(x_i, p_{-i, 1}^{t-1}\right), \forall x_i \in X_i$$ Update the value [TeX:] $$v_{i, 1}^t, \forall i \in V$$ by (11);\ Each player [TeX:] $$i$$ updates the probability of playing strategy 1 by (10): [TeX:] $$p_{i, 1}^t=\left(1-\frac{1}{t}\right) p_{i, 1}^{t-1}+\frac{1}{t} v_{i, 1}^t$$Until: Convergence to a stopping criterion by (12): [TeX:] $$\max \left|P_i^t-P_i^{t-1}\right| \leq \frac{1}{Q}$$ |

it converges to an equilibrium state [TeX:] $$p^*=\left(p_{i, 1}^*, \cdots, p_{n, 1}^*\right)$$ after [TeX:] $$t^*$$ time steps. Then for [TeX:] $$t>t^*$$, the [TeX:] $$P_i^t$$ follows that

where [TeX:] $$\epsilon$$ < 1 is a positive constant.

The following Theorem proves that the equilibrium state is a NE under both the EUT and the PT.

Theorem 2: Consider a covering game G, the presented algorithm in (10) can guarantee that any continuous strategy profile converges to a NE under both the EUT and the PT.

proof : According to (3) and (4), we have

where [TeX:] $$U_i$$(·) can be either the EUT cost or the PT cost.

Define {[TeX:] $$p^t$$} is the presented algorithm process. For the EUT, we can suppose that {[TeX:] $$p^t$$} will converge to an equilibrium state [TeX:] $$p^{\prime}=\left(p_{i, 1}^{\prime}, p_{-i, 1}^{\prime}\right)$$ after time step [TeX:] $$t^{\prime}$$. So, [TeX:] $$v_{i .1}^{t^{\prime}}=v_{i .1}^{t^{\prime}+\beta}, \forall \beta>0$$. I assume that the p is not NE, according to Definition 1, there must exist one player [TeX:] $$i$$ can decrease his cost by changing his continuous strategy.

i) If [TeX:] $$p_{i, 1}^{\prime}$$ = 1, then there exist a [TeX:] $$p_{i, 1} \neq p_{i, 1}^{\prime}$$ such that

[TeX:] $$U_i^{\mathrm{EUT}}\left(p_{i, 1}, p_{-i, 1}^{\prime}\right)<U_i^{\mathrm{EUT}}\left(p_{i, 1}^{\prime}, p_{-i, 1}^{\prime}\right)$$

According to (14), we have

[TeX:] $$\begin{aligned} p_{i, 1}\left[U_i^{\mathrm{EUT}}\right. & \left.\left(1, p_{-i, 1}^{\prime}\right)-U_i^{\mathrm{EUT}}\left(0, p_{-i, 1}^{\prime}\right)\right] \\ & <p_{i, 1}^{\prime}\left[U_i^{\mathrm{EUT}}\left(1, p_{-i, 1}^{\prime}\right)-U_i^{\mathrm{EUT}}\left(0, p_{-i, 1}^{\prime}\right)\right] \end{aligned}$$

it implies

thus, for β > 0, we have [TeX:] $$v_{i, 1}^{t^{\prime}} \neq v_{i, 1}^{t^{\prime}+\beta}$$. ii) If [TeX:] $$p_{i, 1}^{\prime}$$ = 0, similarly, then there exist a [TeX:] $$p_{i, 1} \neq p_{i, 1}^{\prime}$$ such that

then, for β > 0, we have [TeX:] $$v_{i, 1}^{t^{\prime}} \neq v_{i, 1}^{t^{\prime}+\beta}$$.

iii) If [TeX:] $$p_{i, 1}^{\prime} \in(0,1)$$, there exist [TeX:] $$x_i, x_i^{\prime} \in X_i, x_i \neq x_i^{\prime}$$, such that

where [TeX:] $$p_{i, x_i^{\prime}}$$ > 0. Thus, we can choose a value [TeX:] $$\epsilon$$ that satisfies

as [TeX:] $$p^t$$ converges to [TeX:] $$p^{\prime}$$ at time step [TeX:] $$t^{\prime}$$.

Since the presented algorithm process decreases as the number of time steps, the cost distance of a discrete strategy between two neighboring iterations must be less than ε after a certain time steps [TeX:] $$t^{\prime}$$. For t ≥ [TeX:] $$t^{\prime}$$, we have

Thus, for t ≥ [TeX:] $$t^{\prime}$$, the presented algorithm process can be written as

thus, player [TeX:] $$i$$ would not choose the strategy [TeX:] $$x_i^{\prime}$$ but would rather choose strategy [TeX:] $$x_i$$ after [TeX:] $$t^{\prime}$$ time steps. Thus, we have [TeX:] $$p_{i, x_i^{\prime}}$$ = 0. To sum up, the above three cases show that the hypothesis is not hold. Thus the equilibrium state [TeX:] $$p^{\prime}$$ is a NE.

For the PT, since the Prelec function [TeX:] $$w_i$$(·) is continuous and monotonically increasing function over [0,1]. Thus, it is easy to check that an equilibrium state [TeX:] $$p^{\prime \prime}$$ under the PT is also NE. Hence, the proof is completed.

In this paper, the simulation on a variety of networks for illustrating how players’ behaviors impact the overall cover level of networks under both the EUT and the PT, where the networks include the ER random networks [36], the WS small- world networks [37], and the BA scale-free networks [38]. To obtain a NE under both the EUT and the PT, we use the presented algorithm. Moreover, the rational coefficient α is set to 0.5, 0.8 and 1, respectively; the [TeX:] $$Q$$ needs to be set to a large value, i.e., 10000; and the initial continuous strategy profile [TeX:] $$p^0$$ is generated randomly to ensure that the simulation results are not affected by the initial settings.

In this subsection, we consider the three categories of networks with size [TeX:] $$n$$ = 1000. For the WS networks, we consider two cases: WS(I) and WS(II), where the probability [TeX:] $$p_w$$ of rewiring each edge is 0.1 and 0.5, respectively. For the BA network, the probability [TeX:] $$p_b$$ that a new node is connected to an existing node [TeX:] $$i$$ is [TeX:] $$k_i / \sum_j k_j$$. Moreover, the average degree < k > of a network is set to 4. For the convenience of comparison, two indicators are introduced as

In (22), f(p) represents the proportion of covered strategy 1 in the network. In (23), the formula’s first part calculates the number of covered vertices of the network, and the second part penalizes any uncovered edge [TeX:] $$e_{i j}$$ (i.e., [TeX:] $$e_{i j}$$ = 0) with the value [TeX:] $$n$$. The smaller the value [TeX:] $$f(s)$$, the fewer vertices are covered in the network. In particular, if there exists at least one uncovered edge of a network, then we have [TeX:] $$f(s)>n$$; otherwise, [TeX:] $$f(s) \leq n$$.

For each μ, we independently repeat 100 runs of the simulations on each network for each datum, and obtained the average value [TeX:] $$\bar{f}(p)$$ and the average value [TeX:] $$\bar{f}(s)$$. The [TeX:] $$\bar{f}(p)$$ and the [TeX:] $$\bar{f}(s)$$ of different values μ on the several networks are shown in Figs. 2–5. Note that the [TeX:] $$\bar{f}(s)$$ is used to reflect thez cover level of networks.

From Figs. 2–5, we can obtain that

i) Since we can always obtain a stable value by running the proposed algorithm at each run, thus there exists at least one NE under both the EUT and the PT on different networks via the presented algorithm. Moreover, for different networks, the NE are different even if n, < k >, and μ are the same.

ii) Since the value [TeX:] $$\bar{f}(s)$$ is less than [TeX:] $$n$$, thus the vertices’ state under the NE satisfy the vertex cover of network, where the number of covered vertices of the BA network is the least, while that of the WS (I) network is the most.

iii) As penalty coefficient μ increases, the value [TeX:] $$\bar{f}(p)$$ also increases under both the EUT and the PT. For instance, for the ER random network, when we fix α = 0.5 and μ varies from 1.6 to 1.8, the value [TeX:] $$\bar{f}(p)$$ is calculated to grow by 1.65% from 558.01 to 567.21.

iv) More importantly, The NE are different between the PT and the EUT. Compared the EUT, the [TeX:] $$\bar{f}(p)$$ is larger under the PT. More specifically, the smaller the rational coefficient α, the larger the value [TeX:] $$\bar{f}(p)$$. In other words, compared with the EUT, the overall cover probability of the network will increase in which deviate from the rational behavior. Take the WS(I) 1000 < 4 > network for example. Under the PT, when we fixed μ = 1.8, the value [TeX:] $$\bar{f}(p)$$ increases from 652.89 to 659.74 as α varies from 1 to 0.8, and the value [TeX:] $$\bar{f}(p)$$ increases from 659.74 to 675.89 as α varies from 0.8 to 0.5.

v) Moreover, the [TeX:] $$\bar{f}(p)$$ and the [TeX:] $$\bar{f}(s)$$ have the same trend of change, that is, the [TeX:] $$\bar{f}(s)$$ increases as the [TeX:] $$\bar{f}(p)$$ increases.

Finally, we examine the convergence rate with different rational coefficient α. Fig. 6 shows the process of the conver- gence to the equilibrium state on the ER and WS(I) networks under both EUT and PT, respectively, where μ is set to 1.8. Thus, we can see that compared the EUT, the convergence rate of the presented algorithm is slower under the PT. More specifically, the smaller the rational coefficient α, the slower the convergence rate. Here, note that even though the number of time steps required for the players’ strategies to converge is large, the computational requirement of each time step is low. Thus, from our generated results, the convergence time with each α is short. If need, in order to accelerate the convergence rate, a small [TeX:] $$Q$$ is suggested to adopted, but at the expense of obtaining an approximate state rather than an exact NE. In sum, the more serious the deviation from objectivity, the worse the cover level, and the slower the convergence rate of the fictitious play algorithm.

In this paper, we have introduced a game-based approach for studying the vertex cover problem, and explicitly considered the subjectivity of players. In particular, we have developed a covering game model based on prospect theory, where each player subjectively observes his neighbors’ strategies, and determines his strategy in order to minimize the cost under the prospect theory. Then, we have presented a fictitious play distributed algorithm and proved that any state reach a NE under both the EUT and the PT. The simulation result have shown that deviations from objective game-theoretic approach can leads to unexpected results, which depends on the subjective degree of the players. That is, players with a high rational degree will help to obtain wonderful cover results of a network. Future work will focus on vertex cover of networks under the framing effect of the prospect theory [39], [40].

Ke Wu Ke Wu received her B.S. and M.S. degree both from the School of Mathematics and Information Science, Nanchang Hangkong University. She is currently a Lecturer at the School of Science, Guilin University of Aerospace Technology, also at Basic Department, Jiangxi Manufacturing Polytechnic College. Her research interests include traffic planning, game theory and networked optimization.

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