# Default Behaviors of Contractors under Surety Bond in Construction Industry Based on Evolutionary Game Model

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Default Risk in Construction Industry

#### 2.2. Surety Bond in Construction Industry

#### 2.3. Evolutionary Game Analysis in Construction Industry

#### 2.4. Summary of the Literature Review

## 3. Problem Description and Assumptions

#### 3.1. Problem Description

#### 3.2. Model Assumptions

_{of}if the contractors adopt the “not default” strategy, while the revenue is R

_{ou}(R

_{ou}<< R

_{of}) if the contractors default. When the contractors adopt the “default” strategy, the quality of the projects will be reduced and the value of the projects will be much less. So, we can assume R

_{ou}<< R

_{of}. We also suppose the owners’ costs of the construction projects are C

_{of}and C

_{ou}(C

_{ou}< C

_{of}) if the contractors implement the “not default” strategy and “default” strategy, respectively. Here the costs of surety bonds are excluded from C

_{of}and C

_{ou}, and are assumed to be C

_{s}

_{1}and C

_{s}

_{2}under the high-penalty conditional bond and low-penalty unconditional bond. When the contractors implement the “default” strategy, they often reduce their bid price by carrying out unsatisfactory work or using inferior materials to win the bid. So, we can assume that the owners’ costs of construction projects are lower when the contractors implement the “default” strategy—that is, C

_{ou}< C

_{of}. The value L

_{ou}is the owners’ loss resulting from the default of contractors. For the contractors, suppose the revenue from the construction projects is R

_{cf}if they adopt the “not default” strategy and the revenue is R

_{cu}if they adopt the “default” strategy. We further suppose the contractors’ costs of the construction projects are C

_{cf}and C

_{cu}under the “not default” strategy and “default” strategy, respectively. Here, according to Assumption 4 above, we can infer that R

_{cf}− C

_{cf}< R

_{cu}− C

_{cu}. If the contractors adopt a “default” strategy, they will be punished P

_{2}under a low-penalty unconditional bond, while they will not only be punished P

_{1}but also be pursued P

_{3}reimbursement under the high-penalty conditional bond, according to the punishment agreement in the construction contracts. Furthermore, the long-term revenue from the good reputation of surety bond institutions is R

_{cl}and the long-term loss from the poor reputation in surety bond institutions is L

_{cl}. Here, we assume that the non-default behavior will improve the reputation while the default behavior will damage the reputation compared to the situation that they do not conduct the construction projects, and the values of R

_{cl}and L

_{cl}are related to the CMCS promoted by the surety bond system. For both players, ΔP

_{4}is the bid price reduction because of the high-penalty conditional bond. Generally, a composite bid evaluation is implemented to select the contractors under a low-penalty unconditional bond or without a surety bond. However, under a high-penalty conditional bond, the lowest bid price can be selected. Subsequently the bid price will be reduced compared to the composite bid price.

## 4. Static Model

#### 4.1. Static Model Description

#### 4.2. Static Model Analysis

#### 4.2.1. Static Model under Low-Penalty Unconditional Bond

**Proposition**

**1.**

_{1}, y

_{1}), x

_{1}, y

_{1}∈ [0, 1]. The calculation of x

_{1}and y

_{1}is shown as Equations (11) and (12).

**Proof**

**1.**

_{1}, y

_{1}), when x

_{1}, y

_{1}∈ [0, 1]. □

**Proposition**

**2.**

_{cl}+ P

_{2}+ L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, (0, 1) is the ESS of the replicator dynamic system (I) and the strategy is (default, surety bond).

_{cl}+ P

_{2}+ L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, there is no ESS of the replicator dynamic system (I) and there is only a central point (x

_{1}, y

_{1}).

**Proof**

**2.**

_{cl}+ P

_{2}+ L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}or R

_{cl}+ P

_{2}+ L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, the ideal strategies cannot be reached under a low-penalty unconditional bond. Here C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}can be treated as the opportunistic profits of contractors. We assumed that the value of R

_{cl}and L

_{cl}are related to the CMCS promoted by the surety bond system—that is, the value of R

_{cl}and L

_{cl}will be larger if CMCS is better developed. As the value of P

_{2}is relatively smaller than R

_{cl}and L

_{cl}, the relation between R

_{cl}+ P

_{2}+ L

_{cl}and C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}is mainly determined by the value of R

_{cl}and L

_{cl}. Under the low-penalty unconditional bond, if the CMCS is not well-developed (R

_{cl}+ P

_{2}+ L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}), the ESS is (default, surety bond), which is absolutely not a good state. However, CMCS can be developed (R

_{cl}+ P

_{2}+ L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}) to reach a better but not ideal state. However, there is not an ESS in the latter case. So, we will introduce dynamic model analysis in Section 5 to further analyze the situation.

#### 4.2.2. Static Model under High-Penalty Conditional Bond

**Proposition**

**3.**

_{2}, y

_{2}), x

_{2}, y

_{2}∈ [0, 1]. The calculation of x

_{2}and y

_{2}is shown in Equations (18) and (19).

**Proof**

**3.**

_{2}, y

_{2}), when x

_{2}, y

_{2}∈ [0, 1]. □

**Proposition**

**4.**

_{cl}+ P

_{3}+ P

_{1}+ L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}< ΔP

_{4}, (0, 1) is the ESS of the replicator dynamic system (II) and the strategy is (default, surety bond).

_{cl}+ P

_{3}+ P

_{1}+ L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}> ΔP

_{4}, (0, 1) is the ESS of the replicator dynamic system (II) and the strategy is (default, surety bond).

_{cl}+ P

_{3}+ P

_{1}+ L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}< ΔP

_{4}, (1, 1) is the ESS of the replicator dynamic system (II) and the strategy is (not default, surety bond).

_{cl}+ P

_{3}+ P

_{1}+ L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}> ΔP

_{4}, there is no ESS of the replicator dynamic system (II) and there is only a central point (x

_{2}, y

_{2}).

**Proof**

**4.**

_{cl}+ P

_{3}+ P

_{1}+ L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, the strategy (default, surety bond) will be reached (no matter C

_{s}

_{1}< ΔP

_{4}or C

_{s}

_{1}> ΔP

_{4}), which is not our aim. We can improve the state through a better developed CMCS (R

_{cl}+ P

_{3}+ P

_{1}+ L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}) promoted by the surety bond system and a reduced cost of high-penalty conditional bond (C

_{s}

_{1}< ΔP

_{4}), and finally the ideal strategy (not default, surety bond) will thus be reached. However, if the cost of high-penalty conditional bonds cannot be reduced enough (C

_{s}

_{1}> ΔP

_{4}), there is not an ESS point. Thus, we need to further study the situation through dynamic model analysis in the following section.

## 5. Dynamic Model

#### 5.1. Dynamic Model Description

_{cl}, the contractors’ long-term potential revenue from the good reputation of surety bond institutions, is supposed to decrease if more contractors choose the “not default” strategy, while L

_{cl}, the contractors’ long-term potential loss from the poor reputation of surety bond institutions, is supposed to decrease with the growing ratio of contractors implementing the “default” strategy. So, the parameters, R

_{cl}and L

_{cl}, are replaced by dynamic factors R

_{cl}(1 − x) and L

_{cl}∗ x, respectively, where R

_{cl}indicates the upper bound value of contractors’ long-term revenue from the good reputation of surety bond institutions, L

_{cl}indicates the upper bound value of contractors’ long-term loss from the poor reputation of surety bond institutions. We further assume that the cost of surety bonds (C

_{s}

_{1}, C

_{s}

_{2}) and the contractors’ contract penalty for defaults (P

_{1}, P

_{2}) will increase if the proportion of contractors adopting the “not default” strategy decreases, because contractors’ default behavior causes a high risk, which produces the risk premium. So, the parameters, C

_{s}

_{1}(or C

_{s}

_{2}) and P

_{1}(or P

_{2}), are replaced by C

_{s}

_{1}(1 − px) (or C

_{s}

_{2}(1 − qx)) and P

_{1}(1 − mx) (or P

_{2}(1 − nx)), respectively. In detail, p and q describe the influence of contractors’ “not default” strategy on the change of the cost of surety bonds, while m and n describe the influence of contractors’ “not default” strategy on the change of the contract penalty for default. Now C

_{s}

_{1}(or C

_{s}

_{2}) and P

_{1}(or P

_{2}) indicate the upper bound values of surety bond cost and contract penalty for default, respectively. The new calculations of utility and related parameters that we have adjusted are shown in Table 7, Table 8 and Table 9.

#### 5.2. Dynamic Model Analysis

#### 5.2.1. Dynamic Model under Low-penalty Unconditional Bond

**Proposition**

**5.**

_{3}, y

_{3}) and (x

_{4}, 1), x

_{3}, y

_{3}, x

_{4}∈ [0, 1]. The calculation of x

_{3}, y

_{3}and x

_{4}is shown in Equations (22)–(24).

**Proof**

**5.**

_{3}, y

_{3}) and (x

_{4}, 1), when x

_{3}, y

_{3}, x

_{4}∈ [0, 1]. □

**Proposition**

**6.**

_{2}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and P

_{2}(1 − n) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, (0, 1) is the ESS of the replicator dynamic system (III) and the strategy is (default, surety bond).

_{2}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and P

_{2}(1 − n) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, (0, 1) is the ESS of the replicator dynamic system (III) and the strategy is (default, surety bond). If x

_{4}∈ [0, 1], y

_{3}∈ [0, 1], (x

_{3}, y

_{3}) is the central point.

_{2}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and P

_{2}(1 − n) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, (x

_{4}, 1) is the ESS of the replicator dynamic system (III) if x

_{4}∈ [0, 1], y

_{3}∉ [0, 1] and there is also an asymptotic stable point (x

_{3}, y

_{3}) if x

_{4}∈ [0, 1], y

_{3}∈ [0, 1] and L

_{cl}− R

_{cl}− P

_{2n}< 0.

_{2}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and P

_{2}(1 − n) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, there is no ESS of the replicator dynamic system (III). However, there is an asymptotic stable point (x

_{3}, y

_{3}) if x

_{4}∈ [0, 1], y

_{3}∈ [0, 1] and L

_{cl}− R

_{cl}− P

_{2n}< 0.

**Proof**

**6.**

_{cl}, the contractors’ long-term revenue from the good reputation in surety bond institutions, is not large enough (P

_{2}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}), the strategy (default, surety bond) will be reached as ESS under a low-penalty unconditional bond (no matter P

_{2}(1 − n) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}or P

_{2}(1 − n) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}), which is not our goal. So, we need to increase R

_{cl}(P

_{2}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}) to reach a better state—that is, to increase the non-default contractors’ opportunity of getting the bid on construction projects in the future. If P

_{2}(1 − n) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, a good ESS strategy (x

_{4}, 1) and an asymptotic stable point (x

_{3}, y

_{3}) will be reached. Then we decrease the default contractors’ opportunity of getting the bid on construction projects in the future to increase the value of L

_{cl}(P

_{2}(1 − n) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}). However, this scenario has no ESS and still has the asymptotic stable point (x

_{3}, y

_{3}) when x

_{4}∈ [0, 1], y

_{3}∈ [0, 1] and L

_{cl}− R

_{cl}− P

_{2n}< 0. The numerical analysis on the dynamic EG model will be further discussed in Section 6.

#### 5.2.2. Dynamic Model under High-Penalty Conditional Bond

**Proposition**

**7.**

_{5}, y

_{5}) and (x

_{6}, 1), x

_{5}, y

_{5}, x

_{6}∈ [0, 1]. The calculation of x

_{5}, y

_{5}and x

_{6}is shown in Equations (27)–(29).

**Proof**

**7.**

_{5}, y

_{5}) and (x

_{6}, 1), when x

_{5}, y

_{5}, x

_{6}∈ [0, 1]. □

**Proposition**

**8.**

_{3}+ P

_{1}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) < ΔP

_{4}, (0, 1) is the ESS of the replicator dynamic system (IV) and the strategy is (default, surety bond).

_{3}+ P

_{1}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) > ΔP

_{4}, (0, 1) is the ESS of the replicator dynamic system (IV) and the strategy is (default, surety bond).

_{3}+ P

_{1}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) < ΔP

_{4}, (0, 1) and (1, 1) are both the ESS of the replicator dynamic system (IV) and the strategies are (default, surety bond) and (not default, surety bond).

_{3}+ P

_{1}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) > ΔP

_{4}, (0, 1) is the ESS of the replicator dynamic system (IV) and the strategy is (default, surety bond).

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) < ΔP

_{4}, (x

_{6}, 1) is the ESS of the replicator dynamic system (IV) if x

_{6}∈ [0, 1].

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) > ΔP

_{4}, (x

_{6}, 1) is the ESS of the replicator dynamic system (IV) if x

_{6}∈ [0, 1], y

_{5}∉ [0, 1]. There is also an asymptotic stable point (x

_{5}, y

_{5}) if x

_{6}∈ [0, 1], y

_{5}∈ [0, 1].

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) < ΔP

_{4}, (1, 1) is the ESS of the replicator dynamic system (IV) and the strategy is (not default, surety bond).

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s1}(1 − p) > ΔP

_{4}, there is no ESS of the replicator dynamic system (IV). However, there is an asymptotic stable point (x

_{5}, y

_{5}) if L

_{cl}− R

_{cl}− P

_{1m}< 0.

**Proof**

**8.**

_{cl}, and R

_{cl}are not large enough, (P

_{3}+ P

_{1}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}), the strategy (default, surety bond) will be reached as ESS is under a high-penalty conditional bond (no matter C

_{s}

_{1}(1 − p) < ΔP

_{4}or C

_{s}

_{1}(1 − p) > ΔP

_{4}), which is not an ideal state. If the value of L

_{cl}increases when the value of R

_{cl}remains unchanged (P

_{3}+ P

_{1}+ R

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}), the ESS strategy is still not ideal when the cost of the surety bond is not low enough (C

_{s}

_{1}(1 − p) > ΔP

_{4}). However, we can reduce the cost (C

_{s}

_{1}(1 − p) < ΔP

_{4}) to obtain two stable strategies, (default, surety bond) and (not default, surety bond). The latter strategy, (not default, surety bond) is the ideal ESS that we want to reach. If the value of R

_{cl}increases and the value of L

_{cl}remains unchanged (P

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}< C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}), there is a good ESS point (x

_{6}, 1) when x

_{6}∈ [0, 1], y

_{5}∉ [0, 1] and C

_{s}

_{1}(1 − p) > ΔP

_{4}. We do not need to cut the cost of the surety bond (C

_{s}

_{1}(1 − p) < ΔP

_{4}) to reach a better state in this scenario, but we still need to do that when x

_{6}∈ [0, 1], y

_{5}∈ [0, 1] because we only have an asymptotic stable point (x

_{5}, y

_{5}). Furthermore, if the values of L

_{cl}and R

_{cl}both increase (P

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}), the perfect ESS strategy (not default, surety bond) will be reached when C

_{s}

_{1}(1 − p) < ΔP

_{4}. However, once the cost of surety bond is quite large (C

_{s}

_{1}(1 − p) > ΔP

_{4}), there is no ESS point and we only have the asymptotic stable point (x

_{5}, y

_{5}) when L

_{cl}− R

_{cl}− P1m > 0. In the following section, we will further analyze the numerical simulation under a high-penalty conditional bond together with a low-penalty unconditional bond.

## 6. Numerical Simulation

#### 6.1. Case Assumption

_{cl}(the contractors’ long-term loss from the poor reputation in surety bond institutions), R

_{cl}(the contractors’ long-term revenue from the good reputation in surety bond institutions) and C

_{s}

_{1}(or C

_{s}

_{2}) (the owners’ cost of surety bond) on the behaviors between contractors and owners through the numerical simulation on MATLAB software based on the function of ode45. We discuss the evolution path of the game strategy between owners and contractors based on the variables and parameters baseline assumptions shown in Table 12. The values of L

_{cl}, R

_{cl}and C

_{s}

_{1}(or C

_{s}

_{2}) will be set in the part of simulation, Section 6.2.

#### 6.2. Simulation

_{cl}= 200 and L

_{cl}= 400, while for the mature enterprises, we set R

_{cl}= 400 and L

_{cl}= 200. Cases 1 and 2 show the simulation under a low-penalty unconditional bond, and Cases 3 to 6 describe the simulation under a high-penalty conditional bond. The value of C

_{s}

_{1}was set as 100 and 10, which indicates the high cost and low cost of the surety bond, respectively.

_{cl}= 400 and L

_{cl}= 200, with other parameters remaining unchanged, which meets the conditions of P

_{2}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and P

_{2}(1 − n) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}. We chose 100 random initial points to simulate, and we set the starting time to 0 and terminal time to 10. The simulation result of the general evolution tendency is shown as Figure 1a. The point (x

_{3}, y

_{3}) is the asymptotic stable point.

_{cl}= 200 and L

_{cl}= 400, with other parameters remaining unchanged, which meets the conditions of P

_{2}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and P

_{2}(1 − n) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}. We chose 100 random initial points to simulate, and we set the starting time to 0 and terminal time to 10. The simulation result of the general evolution tendency is shown as Figure 1b. The point (x

_{3}, y

_{3}) is the central point.

_{cl}= 400, L

_{cl}= 200 and C

_{s}

_{1}= 10, with other parameters remaining unchanged, which meets the conditions of P

_{3}+ P

_{1}+ L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s}

_{1}(1 − p) < ΔP

_{4}. We chose 100 random initial points to simulate, and we set the starting time to 0 and terminal time to 10. The simulation result of the general evolution tendency is shown as Figure 1c. The point (1, 1) is the ESS point, which means the “not default” strategy and “surety bond” strategy are the equilibrium choices of contractors and owners.

_{cl}= 400, L

_{cl}= 200 and C

_{s}

_{1}= 100, with other parameters remaining unchanged, which meets the conditions of P

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s}

_{1}(1 − p) > ΔP

_{4}. We chose 100 random initial points to simulate, and we set the starting time to 0 and terminal time to 10. The simulation result of the general evolution tendency is shown as Figure 1d. The point (x

_{5}, y

_{5}) is the asymptotic stable point.

_{cl}= 200, L

_{cl}= 400 and C

_{s}

_{1}= 10, with other parameters remaining unchanged, which meets the conditions of P

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s}

_{1}(1 − p) < ΔP

_{4}. We chose 100 random initial points to simulate, and we set the starting time to 0 and terminal time to 10. The simulation result of the general evolution tendency is shown as Figure 1e. The point (1, 1) is the ESS point, which means the “not default” strategy and “surety bond” strategy are the equilibrium choices of contractors and owners.

_{cl}= 200, L

_{cl}= 400 and C

_{s}

_{1}= 100, with other parameters remaining unchanged, which meets the conditions of P

_{3}+ P

_{1}+ R

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}, P

_{3}+ P

_{1}(1 − m) + L

_{cl}> C

_{cf}+ R

_{cu}− R

_{cf}− C

_{cu}and C

_{s}

_{1}(1 − p) > ΔP

_{4}. We chose 100 random initial points to simulate, and we set the starting time to 0 and terminal time to 10. The simulation result of the general evolution tendency is shown as Figure 1f. The point (x

_{5}, y

_{5}) is the central point.

_{cl}, R

_{cl}and C

_{s}

_{1}(or C

_{s}

_{2}) on the behaviors between owners and contractors will be further analyzed.

#### 6.3. Influence Analysis of Parameters

_{cl}= 0, C

_{s}

_{2}= 50, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 2 show the results of the sensitivity analysis, in which R

_{cl}increases from 0 to 320 with an addition of 80. From panel (a), we can see that with an increasing R

_{cl}, contractors are more likely to change its strategy from “default” to “not default”. The more loss caused by the poor reputation in surety bond institutions will help restrict the default behavior of contractors. For panel (b), if R

_{cl}increases, the proportion of owners implementing the “surety bond” strategy will decrease.

_{cl}= 0, C

_{s}

_{2}= 50, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 3 show the results of the sensitivity analysis, in which L

_{cl}increases from 0 to 320 with an addition of 80. From the results, we can see that the changes of L

_{cl}have almost no influence on the behaviors between owners and contractors.

_{cl}= 0, L

_{cl}= 0, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 4 show the results of the sensitivity analysis, in which C

_{s}

_{2}increases from 10 to 90 with an addition of 40. From panel (b), when C

_{s}

_{2}increases, the strategies of owners and contractors do not change, but the owners reach the surety bond strategy much more slowly.

_{cl}= 400, L

_{cl}= 200, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 5 show the results of the sensitivity analysis, in which C

_{s}

_{2}increases from 10 to 90 with an addition of 40. From panels (a) and (b), the increasing C

_{s}

_{2}has a great influence on the strategies of owners and contractors. More contractors are likely to change from the “not default” to “default” strategy, while the owners change from the surety bond to the “not surety bond” strategy.

_{cl}= 0, C

_{s}

_{1}= 70, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 6 show the results of the sensitivity analysis, in which R

_{cl}increases from 0 to 400 with an addition of 100. In panel (a), it can be seen that with the increasing R

_{cl}, more contractors are likely to change their strategy from “default” to “not default”.

_{cl}= 0, C

_{s}

_{1}= 10, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 7 show the results of the sensitivity analysis, in which R

_{cl}increases from 0 to 400 with an addition of 100. From panel (a), we can see that with the increasing R

_{cl}, more contractors are likely to change their strategy from “default” to “not default”.

_{cl}= 0, C

_{s}

_{1}= 70, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 8 show the results of the sensitivity analysis, in which L

_{cl}increases from 0 to 400 with an addition of 100. From panel (a), we can see that when L

_{cl}increases, more contractors are likely to change the “default” strategy to the “not default” strategy. However, the rate of convergence decreases with an increasing L

_{cl}.

_{cl}= 400, C

_{s}

_{1}= 70, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 9 show the results of the sensitivity analysis, in which L

_{cl}increases from 0 to 400 with an addition of 100. In panel (a), it can be seen that if L

_{cl}increases, the proportion of contractors adopting the “not default” strategy will also increase. Note that the rate of convergence decreases with the increasing L

_{cl}.

_{cl}= 0, C

_{s}

_{1}= 10, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 10 show the results of the sensitivity analysis, in which L

_{cl}increases from 0 to 400 with an addition of 100. In panel (a), more contractors are likely to implement the “not default” strategy with the increasing L

_{cl}.

_{cl}= 0, L

_{cl}= 0, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 11 show the results of the sensitivity analysis, in which C

_{s}

_{1}increases from 10 to 70 with an addition of 30. From the results, we can see that the changes of C

_{s}

_{1}have almost no influence on the behaviors between owners and contractors.

_{cl}= 400, L

_{cl}= 200, and keeping the other parameters consistent with the baseline, the initial point is (0.4, 0.4). Panels (a) and (b) in Figure 12 show the results of the sensitivity analysis, in which C

_{s}

_{1}increases from 10 to 70 with an addition of 30. We can see from both panels that the increasing C

_{s}

_{1}will reduce the rate of convergence and change the strategies of owners and contractors. The proportion of contractors implementing the “not default” strategy and the ratio of owners implementing the “surety bond” strategy both decrease with an increasing C

_{s}

_{1}.

## 7. Discussions

- 1.
- Low-penalty unconditional bond

_{cl}plays a leading role in the influence of the contractors’ strategy choice. More contractors are likely to adopt the “not default” strategy with an increasing R

_{cl}. C

_{s}

_{2}is also an important factor that influences the contractors’ behavior. When R

_{cl}and L

_{cl}are large enough, the decreasing C

_{s}

_{2}will help change the contractors’ strategy from the “default” one to “not default” one. However, if R

_{cl}and L

_{cl}are not large enough, C

_{s}

_{2}has almost no influence on the contractors’ behavior. For the factor L

_{cl}, it has no effect on the strategy choice of contractors, unless combined with R

_{cl}and L

_{cl}.

- 2.
- High-penalty conditional bond

_{cl}still plays a leading role in the influence of the contractors’ strategy choice under a high-penalty conditional bond. With the value of R

_{cl}increasing, more contractors tend to implement the “not default” strategy. Similar to the situation under a low-penalty unconditional bond, C

_{s}

_{1}is also important to influence the contractors’ behavior when R

_{cl}and L

_{cl}are large enough, as the decreasing C

_{s}

_{1}will help the contractors to implement the “not default” strategy with a better rate of convergence. However, different from the situation under a low-penalty unconditional bond, the factor L

_{cl}produces a positive effect on the contractors’ behavior. More contractors want to adopt the “not default” strategy when L

_{cl}increases. However, if the difference between R

_{cl}and L

_{cl}is reduced, the rate of convergence will also decrease.

- 3.
- Comparison

_{cl}has a more positive effect on the contractors’ behavior under a high-penalty conditional bond. The effect of the cost of surety bonds is almost the same under these two kinds of surety bond, but generally the lower cost of a high-penalty conditional bond will produce more positive effects. The factor L

_{cl}influences the contractors’ behavior positively under a high-penalty conditional bond, while it has almost no influence under a low-penalty unconditional bond.

_{cl}and L

_{cl}are positively determined by the development of CMCS, and we should further develop the CMCS to enlarge the values of R

_{cl}and L

_{cl}. Moreover, we need to keep the value of R

_{cl}a little larger than L

_{cl}to reach a more efficient system according to the discussion above. That is to say, the long-term revenue of the “not default” strategy should exceed the long-term loss of the “default” strategy. Furthermore, a larger gap between R

_{cl}and L

_{cl}will help the market to reach the ideal state more quickly. Second, reducing the cost of surety bonds is helpful to the contractors’ “not default” strategy choice only when the CMCS has been established as driven by surety bonds. If the CMCS is not developed well, the cost of surety bonds almost produces no effects on the contractors’ strategy. As a result, CMCS should first be established, and then surety bond institutions need to be subsidized to reduce the cost at the early stage. After the surety bond system has been well developed, some discounts can be provided for the companies with a good reputation.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

- Chava, S.; Jarrow, R.A. Bankruptcy prediction with industry effects. Rev. Finance
**2004**, 8, 537–569. [Google Scholar] [CrossRef] - Tserng, H.P.; Lin, G.-F.; Tsai, L.K.; Chen, P.-C. An enforced support vector machine model for construction contractor default prediction. Autom. Constr.
**2011**, 20, 1242–1249. [Google Scholar] [CrossRef] - Cruz, C.O.; Marques, R.C. Using probabilistic methods to estimate the public sector comparator. Comput. Civ. Infrastruct. Eng.
**2012**, 27, 782–800. [Google Scholar] [CrossRef] - Horta, I.M.; Camanho, A.S. Company failure prediction in the construction industry. Expert Syst. Appl.
**2013**, 40, 6253–6257. [Google Scholar] [CrossRef] - Tserng, H.P.; Ngo, T.L.; Chen, P.C.; Tran, L.Q. A grey system theory-based default prediction model for construction firms. Comput. Civ. Infrastruct. Eng.
**2014**, 30, 120–134. [Google Scholar] [CrossRef] - Wu, W.-W. Beyond business failure prediction. Expert Syst. Appl.
**2010**, 37, 2371–2376. [Google Scholar] [CrossRef] - Kangari, R.; Farid, F.; Elgharib, H.M. Financial performance analysis for construction industry. J. Constr. Eng. Manag.
**1992**, 118, 349–361. [Google Scholar] [CrossRef] - Russell, J.S.; Zhai, H. Predicting contractor failure using stochastic dynamics of economic and financial variables. J. Constr. Eng. Manag.
**1996**, 122, 183–191. [Google Scholar] [CrossRef] [Green Version] - Ng, S.T.; Wong, J.M.; Zhang, J. Applying Z-score model to distinguish insolvent construction companies in China. Habitat Int.
**2011**, 35, 599–607. [Google Scholar] [CrossRef] - Tserng, H.P.; Chen, P.-C.; Huang, W.-H.; Lei, M.C.; Tran, Q.H. Prediction of default probability for construction firms using the logit model. J. Civ. Eng. Manag.
**2014**, 20, 247–255. [Google Scholar] [CrossRef] [Green Version] - Cheng, M.-Y.; Hoang, N.-D.; Limanto, L.; Wu, Y.-W. A novel hybrid intelligent approach for contractor default status prediction. Knowl.-Based Syst.
**2014**, 71, 314–321. [Google Scholar] [CrossRef] - Choi, H.; Son, H.; Kim, C. Predicting financial distress of contractors in the construction industry using ensemble learning. Expert Syst. Appl.
**2018**, 110, 1–10. [Google Scholar] [CrossRef] - You, J.; Chen, Y.; Wang, W.; Shi, C. Uncertainty, opportunistic behavior, and governance in construction projects: The efficacy of contracts. Int. J. Proj. Manag.
**2018**, 36, 795–807. [Google Scholar] [CrossRef] - Nasir, M.K.; Hadikusumo, B.H.W. System dynamics model of contractual relationships between owner and contractor in construction projects. J. Manag. Eng.
**2019**, 35, 04018052. [Google Scholar] [CrossRef] - Zhang, L.; Qian, Q. How mediated power affects opportunism in owner–contractor relationships: The role of risk perceptions. Int. J. Proj. Manag.
**2017**, 35, 516–529. [Google Scholar] [CrossRef] - Liu, J.; Zhao, X.; Li, Y. Exploring the factors inducing contractors’ unethical behavior: Case of China. J. Prof. Issues Eng. Educ. Pr.
**2017**, 143, 04016023. [Google Scholar] [CrossRef] - Lu, W.; Zhang, L.; Zhang, L. Effect of contract completeness on contractors’ opportunistic behavior and the moderating role of interdependence. J. Constr. Eng. Manag.
**2016**, 142, 04016004. [Google Scholar] [CrossRef] - Zhang, S.; Zhang, S.; Gao, Y.; Ding, X. Contractual governance: Effects of risk allocation on contractors’ cooperative behavior in construction projects. J. Constr. Eng. Manag.
**2016**, 142, 04016005. [Google Scholar] [CrossRef] - Kangari, R.; Bakheet, M. Construction surety bonding. J. Constr. Eng. Manag.
**2001**, 127, 232–238. [Google Scholar] [CrossRef] - Awad, A.; Fayek, A.R. Adaptive learning of contractor default prediction model for surety bonding. J. Constr. Eng. Manag.
**2013**, 139, 694–704. [Google Scholar] [CrossRef] - Deng, X.; Ding, S.; Tian, Q. Reasons underlying a mandatory high penalty construction contract bonding system. J. Constr. Eng. Manag.
**2004**, 130, 67–74. [Google Scholar] [CrossRef] - Eaglestone, F.N.; Smyth, C. Insurance under the ICE Contract; Godwin: London, UK, 1985. [Google Scholar]
- Russell, J.S. Surety Bonds for Construction Contracts; American Society of Civil Engineers (ASCE): Reston, VA, USA, 2000. [Google Scholar]
- Winston, S.; Ichniowski, T. Procurement: Miller act reforms approved by house. Eng. Newsrev.
**1999**, 243, 11. [Google Scholar] - Bayraktar, M.E.; Hastak, M. Scoring approach to construction bond underwriting. J. Constr. Eng. Manag.
**2010**, 136, 957–967. [Google Scholar] [CrossRef] - Marsh, K.; Fayek, A.R. SuretyAssist: Fuzzy expert system to assist surety underwriters in evaluating construction contractors for bonding. J. Constr. Eng. Manag.
**2010**, 136, 1219–1226. [Google Scholar] [CrossRef] - Awad, A.; Fayek, A.R. A decision support system for contractor prequalification for surety bonding. Autom. Constr.
**2012**, 21, 89–98. [Google Scholar] [CrossRef] - El-Mashaleh, M.S.; Horta, I.M. Evaluating contractors for bonding: DEA decision making model for surety underwriters. J. Manag. Eng.
**2016**, 32, 04015020. [Google Scholar] [CrossRef] - Al-Sobiei, O.S.; Arditi, D.; Polat, G. Managing owner’s risk of contractor default. J. Constr. Eng. Manag.
**2005**, 131, 973–978. [Google Scholar] [CrossRef] - Nash, J. Non-cooperative games. Ann. Math.
**1951**, 54, 286–295. [Google Scholar] [CrossRef] - Smith, J.M.; Price, G.R. The logic of animal conflict. Nat. Cell Biol.
**1973**, 246, 15–18. [Google Scholar] [CrossRef] - Smith, J.M. The theory of games and the evolution of animal conflicts. J. Theor. Biol.
**1974**, 47, 209–221. [Google Scholar] [CrossRef] [Green Version] - Taylor, P.D.; Jonker, L.B. Evolutionary stable strategies and game dynamics. Math. Biosci.
**1978**, 40, 145–156. [Google Scholar] [CrossRef] - Bester, H.; Güth, W. Is altruism evolutionarily stable? J. Econ. Behav. Organ.
**1998**, 34, 193–209. [Google Scholar] [CrossRef] [Green Version] - Dufwenberg, M.; Güth, W. Indirect evolution vs. strategic delegation: A comparison of two approaches to explaining economic institutions. Eur. J. Political-Econ.
**1999**, 15, 281–295. [Google Scholar] [CrossRef] - Naini, S.G.J.; Aliahmadi, A.R.; Jafari-Eskandari, M. Designing a mixed performance measurement system for environmental supply chain management using evolutionary game theory and balanced scorecard: A case study of an auto industry supply chain. Resour. Conserv. Recycl.
**2011**, 55, 593–603. [Google Scholar] [CrossRef] - Wang, G.; Xue, Y.; Skibniewski, M.J.; Song, J.; Lu, H. Analysis of private investors conduct strategies by governments supervising public-private partnership projects in the new media era. Sustainability
**2018**, 10, 4723. [Google Scholar] [CrossRef] [Green Version] - Li, L.; Li, Z.; Jiang, L.; Wu, G.; Cheng, D. Enhanced cooperation among stakeholders in PPP mega-infrastructure projects: A China study. Sustainability
**2018**, 10, 2791. [Google Scholar] [CrossRef] [Green Version] - Zhu, J.; Fang, M.; Shi, Q.; Wang, P.; Li, Q. Contractor cooperation mechanism and evolution of the green supply chain in mega projects. Sustainability
**2018**, 10, 4306. [Google Scholar] [CrossRef] [Green Version] - Hao, C.; Du, Q.; Huang, Y.; Shao, L.; Yan, Y. Evolutionary game analysis on knowledge-sharing behavior in the construction supply chain. Sustainability
**2019**, 11, 5319. [Google Scholar] [CrossRef] [Green Version] - Yang, Y.; Tang, W.; Shen, W.; Wang, T. Enhancing risk management by partnering in international EPC projects: Perspective from evolutionary game in chinese construction companies. Sustainability
**2019**, 11, 5332. [Google Scholar] [CrossRef] [Green Version] - Zheng, L.; Lu, W.; Chen, K.; Chau, K.W.; Niu, Y. Benefit sharing for BIM implementation: Tackling the moral hazard dilemma in inter-firm cooperation. Int. J. Proj. Manag.
**2017**, 35, 393–405. [Google Scholar] [CrossRef] [Green Version] - Du, Y.; Zhou, H.; Yuan, Y.; Xue, H. Exploring the moral hazard evolutionary mechanism for bim implementation in an integrated project team. Sustainability
**2019**, 11, 5719. [Google Scholar] [CrossRef] [Green Version] - Jide, S.; Xincheng, W.; Liangfa, S. Research on the mobility behaviour of Chinese construction workers based on evolutionary game theory. Econ. Res.
**2018**, 31, 1–14. [Google Scholar] [CrossRef] - Pi, Z.; Gao, X.; Chen, L.; Liu, J. The new path to improve construction safety performance in China: An evolutionary game theoretic approach. Int. J. Environ. Res. Public Heal.
**2019**, 16, 2443. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, Y.; Zhu, D.; Zhou, L. A game theory analysis of promoting the spongy city construction at the building and community scale. Habitat Int.
**2019**, 86, 91–100. [Google Scholar] [CrossRef] - Chen, J.; Hua, C.; Liu, C. Considerations for better construction and demolition waste management: Identifying the decision behaviors of contractors and government departments through a game theory decision-making model. J. Clean. Prod.
**2019**, 212, 190–199. [Google Scholar] [CrossRef] - Jide, S.; Xincheng, W.; Liangfa, S. Chinese construction workers’ behaviour towards attending vocational skills trainings: Evolutionary game theory with government participation. J. Differ. Equ. Appl.
**2016**, 23, 468–485. [Google Scholar] [CrossRef] - Shi, Q.; Zhu, J.; Li, Q. Cooperative evolutionary game and applications in construction supplier tendency. Complexity
**2018**, 2018, 8401813. [Google Scholar] [CrossRef] - Xu, X.; Li, Z.; Wang, J.; Huang, W. Collaboration between designers and contractors to improve building energy performance. J. Clean. Prod.
**2019**, 219, 20–32. [Google Scholar] [CrossRef] - Pan, Y.; Deng, X.; Maqbool, R.; Niu, W. Insurance crisis, legal environment, and the sustainability of professional liability insurance market in the construction industry: Based on the US market. Adv. Civ. Eng.
**2019**, 2019, 1614868. [Google Scholar] [CrossRef] [Green Version] - Friedman, D. Evolutionary games in economics. Econ. J. Econ. Soc.
**1991**, 59, 637. [Google Scholar] [CrossRef] [Green Version] - Chen, W.; Hu, Z.-H. Using evolutionary game theory to study governments and manufacturers’ behavioral strategies under various carbon taxes and subsidies. J. Clean. Prod.
**2018**, 201, 123–141. [Google Scholar] [CrossRef] - Fang, Y.; Chen, L.; Mei, S.; Wei, W.; Huang, S.; Liu, F. Coal or electricity? An evolutionary game approach to investigate fuel choices of urban heat supply systems. Energy
**2019**, 181, 107–122. [Google Scholar] [CrossRef]

**Figure 1.**Evolution paths of Case 1 to Case 6. (

**a**) Case 1. (

**b**) Case 2. (

**c**) Case 3. (

**d**) Case 4. (

**e**) Case 5. (

**f**) Case 6.

**Figure 4.**Influence of C

_{s}

_{2}in Case 1.3. (

**a**) Influence on contractors. (

**b**) Influence on owners.

**Figure 5.**Influence of C

_{s}

_{2}in Case 1.4. (

**a**) Influence on contractors. (

**b**) Influence on owners.

**Figure 11.**Influence of C

_{s}

_{1}in Case 2.6. (

**a**) Influence on contractors. (

**b**) Influence on owners.

**Figure 12.**Influence of C

_{s}

_{1}in Case 2.7. (

**a**) Influence on contractors. (

**b**) Influence on owners.

Variables | Descriptions | Variables | Descriptions |

x | The proportion of contractors adopting “not default” strategy | y | The proportion of owners adopting “surety bond” strategy |

Parameters | Descriptions | Parameters | Descriptions |

R_{of} | The owners’ revenue from the construction projects if the contractors adopt “not default” strategy | C_{of} | The owners’ cost on the construction projects if the contractors adopt “not default” strategy |

R_{ou} | The owners’ revenue from the construction projects if the contractors adopt “default” strategy | C_{ou} | The owners’ cost on the construction projects if the contractors adopt “default” strategy |

R_{cf} | The contractors’ revenue from the construction projects if they adopt “not default” strategy | C_{cf} | The contractors’ cost on the construction projects if they adopt “not default” strategy |

R_{cu} | The contractors’ revenue from the construction projects if they adopt “default” strategy | C_{cu} | The contractors’ cost on the construction projects if they adopt “default” strategy |

R_{cl} | The contractors’ long-term revenue from the good reputation in surety bond institutions | L_{cl} | The contractors’ long-term loss from the poor reputation in surety bond institutions |

P_{1} | The contractors’ punishment for default under high-penalty conditional bond | L_{ou} | The owners’ loss resulting from the default of the contractors |

P_{2} | The contractors’ punishment for default under low-penalty unconditional bonds | C_{s}_{1} | The owners’ cost on the high-penalty conditional bond |

P_{3} | The contractors’ reimbursement due to default under high-penalty conditional bond | C_{s}_{2} | The owners’ cost on the low-penalty unconditional bond |

ΔP_{4} | The bid price reduction under high-penalty conditional bond |

Contractors | Owners | |
---|---|---|

Surety Bond (y) | Not Surety Bond (1 − y) | |

not default (x) | C_{11}, O_{11} | C_{12}, O_{12} |

default (1 − x) | C_{21}, O_{21} | C_{22}, O_{22} |

**Table 3.**Calculation of the utility under low-penalty unconditional bond in payoff matrix in static model.

Utility | Calculation | Utility | Calculation |
---|---|---|---|

C_{11} | R_{cf} + R_{cl} − C_{cf} | O_{11} | R_{of} − C_{of} − C_{s}_{2} |

C_{12} | R_{cf} − C_{cf} | O_{12} | R_{of} − C_{of} |

C_{21} | R_{cu} − C_{cu} − P_{2}− L_{cl} | O_{21} | R_{ou} + P_{2} − C_{ou} − C_{s}_{2} − L_{ou} |

C_{22} | R_{cu} − C_{cu} | O_{22} | R_{ou} − C_{ou} − L_{ou} |

**Table 4.**Calculation of the utility under a high-penalty conditional bond in payoff matrix in static model.

Variables | Descriptions | Variables | Descriptions |
---|---|---|---|

C_{11} | R_{cf} + R_{cl} − C_{cf} − ΔP_{4} | O_{11} | R_{of} − C_{of} − C_{s}_{1}+ ΔP_{4} |

C_{12} | R_{cf} − C_{cf} | O_{12} | R_{of} − C_{of} |

C_{21} | R_{cu} − C_{cu} − P_{3}− P_{1}− L_{cl} − ΔP_{4} | O_{21} | R_{of} + P_{1}− C_{ou} − C_{s}_{1}+ ΔP_{4} |

C_{22} | R_{cu} − C_{cu} | O_{22} | R_{ou} − C_{ou} − L_{ou} |

Equilibrium Points | R_{cl} + P_{2}+ L_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu} | R_{cl} + P_{2}+ L_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||
---|---|---|---|---|---|---|

detJ1 | trJ1 | State | detJ1 | trJ1 | State | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point |

(0, 1) | + | − | ESS | − | ± | Saddle point |

(1, 0) | − | ± | Saddle point | − | ± | Saddle point |

(1, 1) | + | + | Instability point | − | ± | Saddle point |

(x_{1}, y_{1}) | / | / | / | + | 0 | Central point |

Equilibrium Points | R_{cl} + P_{3}+ P_{1}+ L_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu} | |||||

C_{s}_{1}< ΔP_{4} | C_{s}_{1}> ΔP_{4} | |||||

detJ2 | trJ2 | State | detJ2 | trJ2 | State | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point |

(0, 1) | + | − | ESS | + | − | ESS |

(1, 0) | + | + | Instability point | − | ± | Saddle point |

(1, 1) | − | ± | Saddle point | + | + | Instability point |

Equilibrium Points | R_{cl} + P_{3}+ P_{1}+ L_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu} | |||||

C_{s}_{1}< ΔP_{4} | C_{s}_{1}> ΔP_{4} | |||||

detJ2 | trJ2 | State | detJ2 | trJ2 | State | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point |

(0, 1) | − | ± | Saddle point | − | ± | Saddle point |

(1, 0) | + | + | Instability point | − | ± | Saddle point |

(1, 1) | + | − | ESS | − | ± | Saddle point |

(x_{2}, y_{2}) | / | / | / | + | 0 | Central point |

**Table 7.**Calculation of the utility under low-penalty unconditional bond in payoff matrix in dynamic model.

Utility | Calculation | Utility | Calculation |
---|---|---|---|

C_{11} | R_{cf} + R_{cl}(1 − x) − C_{cf} | O_{11} | R_{of} − C_{of} − C_{s}_{2}(1 − qx) |

C_{12} | R_{cf} − C_{cf} | O_{12} | R_{of} − C_{of} |

C_{21} | R_{cu} − C_{cu} − P_{2}(1 − nx) − L_{cl} ∗ x | O_{21} | R_{ou} + P_{2}(1 − nx) − C_{ou} − C_{s}_{2}(1 − qx) − L_{ou} |

C_{22} | R_{cu} − C_{cu} | O_{22} | R_{ou} − C_{ou} − L_{ou} |

**Table 8.**Calculation of the utility under high-penalty conditional bond in payoff matrix in dynamic model.

Variables | Descriptions | Variables | Descriptions |
---|---|---|---|

C_{11} | R_{cf} + R_{cl}(1 − x) − C_{cf} − ΔP_{4} | O_{11} | R_{of} −C_{of} −C_{s}_{1}(1 − px) + ΔP_{4} |

C_{12} | R_{cf} −C_{cf} | O_{12} | R_{of} −C_{of} |

C_{21} | R_{cu} −C_{cu} −P_{3}−P_{1}(1 − mx) − L_{cl}*x − ΔP_{4} | O_{21} | R_{of} + P_{1}(1 − mx) − C_{ou} − C_{s}_{1}(1 − px) + ΔP_{4} |

C_{22} | R_{cu} −C_{cu} | O_{22} | R_{ou} −C_{ou} −L_{ou} |

Variables | Descriptions | Variables | Descriptions |

x | The proportion of contractors adopting “not default” strategy | y | The proportion of owners adopting “surety bond” strategy |

Parameters | Descriptions | Parameters | Descriptions |

R_{of} | The owners’ revenue from the construction projects if the contractors adopt “not default” strategy | C_{of} | The owners’ cost on the construction projects if the contractors adopt “not default” strategy |

R_{ou} | The owners’ revenue from the construction projects if the contractors adopt “default” strategy | C_{ou} | The owners’ cost on the construction projects if the contractors adopt “default” strategy |

R_{cf} | The contractors’ revenue from the construction projects if they adopt “not default” strategy | C_{cf} | The contractors’ cost on the construction projects if they adopt “not default” strategy |

R_{cu} | The contractors’ revenue from the construction projects if they adopt “default” strategy | C_{cu} | The contractors’ cost on the construction projects if they adopt “default” strategy |

R_{cl}(1 − x) | The contractors’ dynamic long-term revenue from the good reputation in surety bond institutions | L_{cl} ∗ x | The contractors’ dynamic long-term loss from the poor reputation in surety bond institutions |

P_{1}(1 − mx), m ∈ [0, 1] | The contractors’ dynamic punishment for default under high-penalty conditional bond | L_{ou} | The owners’ loss resulting from the default of the contractors |

P_{2}(1 − nx), n ∈ [0, 1] | The contractors’ dynamic punishment for default under low-penalty unconditional bond | C_{s}_{1}(1 − px), p ∈ [0, 1] | The owners’ dynamic cost on the high-penalty conditional bond |

P_{3} | The contractors’ reimbursement due to default under high-penalty conditional bond | C_{s}_{2}(1 − qx), q ∈ [0, 1] | The owners’ dynamic cost on the low-penalty unconditional bond |

ΔP_{4} | The bid price reduction under high-penalty conditional bond |

**Table 10.**Stability of the equilibrium points in dynamic model under low-penalty unconditional bond.

Equilibrium Points | P_{2}+ R_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

P_{2}(1 − n) + L_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu} | P_{2}(1 − n) + L_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

detJ3 | trJ3 | State | detJ3 | trJ3 | State | Special Condition | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(0, 1) | + | − | ESS | + | − | ESS | / |

(1, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(1, 1) | + | + | Instability point | − | ± | Saddle point | / |

(x_{3}, y_{3}) | / | / | / | + | + | Central point | x_{4} ∈ [0, 1]y _{3} ∈ [0, 1] |

(x_{4}, 1) | / | / | / | − | ± | Saddle point | x_{4} ∈ [0, 1]y _{3} ∈ [0, 1] |

(x_{4}, 1) | / | / | / | + | + | Instability point | x_{4} ∈ [0, 1]y _{3} ∉ [0, 1] |

Equilibrium Points | P_{2}+ R_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

P_{2}(1 − n) + L_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu} | P_{2}(1 − n) + L_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

detJ3 | trJ3 | State | detJ3 | trJ3 | State | Special Condition | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(0, 1) | − | ± | Saddle point | − | ± | Saddle point | / |

(1, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(1, 1) | + | + | Instability point | − | ± | Saddle point | / |

(x_{3}, y_{3}) | + | − | Asymptotic stable point | + | − | Asymptotic stable point | x_{4} ∈ [0, 1]y _{3} ∈ [0, 1]L _{cl} − R_{cl2} − Pn < 0 |

(x_{3}, y_{3}) | / | / | / | + | + | Central point | x_{4} ∈ [0, 1]y _{3} ∈ [0, 1]L _{cl} − R_{cl 2} − Pn > 0 |

(x_{4}, 1) | − | ± | Saddle point | / | / | / | x_{4} ∈ [0, 1]y _{3} ∈ [0, 1] |

(x_{4}, 1) | + | − | ESS | / | / | / | x_{4} ∈ [0, 1]y _{3} ∉ [0, 1] |

Equilibrium Points | P_{3}+ P_{1}+ R_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu}P_{3}+ P_{1}(1 − m) + L_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

C_{s}_{1}(1 − p) < ΔP_{4} | C_{s}_{1}(1 − p) > ΔP_{4} | ||||||

detJ4 | trJ4 | State | detJ4 | trJ4 | State | Special Condition | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(0, 1) | + | − | ESS | + | − | ESS | / |

(1, 0) | + | + | Instability point | − | ± | Saddle point | / |

(1, 1) | − | ± | Saddle point | + | + | Instability point | / |

Equilibrium Points | P_{3}+ P_{1}+ R_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu}P_{3}+ P_{1}(1 − m) + L_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

C_{s}_{1}(1 − p) < ΔP_{4} | C_{s}_{1}(1 − p) > ΔP_{4} | ||||||

detJ4 | trJ4 | State | detJ4 | trJ4 | State | Special Condition | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(0, 1) | + | − | ESS | + | − | ESS | / |

(1, 0) | + | + | Instability point | - | ± | Saddle point | / |

(1, 1) | + | − | ESS | − | ± | Saddle point | / |

(x_{5}, y_{5}) | / | / | / | + | + | Central point | x_{6} ∈ [0, 1]y _{5} ∈ [0, 1] |

(x_{6}, 1) | − | ± | Saddle point | − | ± | Saddle point | x_{6} ∈ [0, 1]y _{5} ∈ [0, 1] |

(x_{6}, 1) | − | ± | Saddle point | + | + | Instability point | x_{6} ∈ [0, 1]y _{5} ∉ [0, 1] |

Equilibrium Points | P_{3}+ P_{1}+ R_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu}P_{3}+ P_{1}(1 − m) + L_{cl} < C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

C_{s}_{1}(1 − p) < ΔP_{4} | C_{s}_{1}(1 − p) > ΔP_{4} | ||||||

detJ4 | trJ4 | State | detJ4 | trJ4 | State | Special Condition | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(0, 1) | − | ± | Saddle point | − | ± | Saddle point | / |

(1, 0) | + | + | Instability point | − | ± | Saddle point | / |

(1, 1) | − | ± | Saddle point | + | + | Instability point | / |

(x_{5}, y_{5}) | / | / | / | + | − | Asymptotic stable point | x_{6} ∈ [0, 1]y _{5} ∈ [0, 1] |

(x_{6}, 1) | + | − | ESS | − | ± | Saddle point | x_{6} ∈ [0, 1]y _{5} ∈ [0, 1] |

(x_{6}, 1) | + | − | ESS | + | − | ESS | x_{6} ∈ [0, 1]y _{5} ∉ [0, 1] |

Equilibrium Points | P_{3}+ P_{1}+ R_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu}P_{3}+ P_{1}(1 − m) + L_{cl} > C_{cf} + R_{cu} − R_{cf} − C_{cu} | ||||||

C_{s}_{1}(1 − p) < ΔP_{4} | C_{s}_{1}(1 − p) > ΔP_{4} | ||||||

detJ4 | trJ4 | State | detJ4 | trJ4 | State | Special Condition | |

(0, 0) | − | ± | Saddle point | − | ± | Saddle point | / |

(0, 1) | − | ± | Saddle point | − | ± | Saddle point | / |

(1, 0) | + | + | Instability point | − | ± | Saddle point | / |

(1, 1) | + | − | ESS | − | ± | Saddle point | / |

(x_{5}, y_{5}) | / | / | / | + | − | Asymptotic stable point | Lcl − Rcl − P_{1m} < 0 |

(x_{5}, y_{5}) | / | / | / | + | + | Central point | Lcl − Rcl − P_{1m} > 0 |

Variables | Values (Million Dollars) | Variables | Values (Million Dollars) |

R_{cf} | 1000 | C_{cf} | 900 |

R_{cu} | 900 | C_{cu} | 600 |

R_{of} | 1100 | P_{3} | 150 |

R_{ou} | 300 | L_{ou} | 200 |

ΔP_{4} | 30 | P_{1} | 30 |

P_{2} | 110 | C_{s}_{2} | 50 |

Parameters | Values | Parameters | Values |

p | 0.3 | m | 0.5 |

q | 0.3 | n | 0.5 |

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## Share and Cite

**MDPI and ACS Style**

Jing, J.; Deng, X.; Maqbool, R.; Rashid, Y.; Ashfaq, S.
Default Behaviors of Contractors under Surety Bond in Construction Industry Based on Evolutionary Game Model. *Sustainability* **2020**, *12*, 9162.
https://doi.org/10.3390/su12219162

**AMA Style**

Jing J, Deng X, Maqbool R, Rashid Y, Ashfaq S.
Default Behaviors of Contractors under Surety Bond in Construction Industry Based on Evolutionary Game Model. *Sustainability*. 2020; 12(21):9162.
https://doi.org/10.3390/su12219162

**Chicago/Turabian Style**

Jing, Jiabao, Xiaomei Deng, Rashid Maqbool, Yahya Rashid, and Saleha Ashfaq.
2020. "Default Behaviors of Contractors under Surety Bond in Construction Industry Based on Evolutionary Game Model" *Sustainability* 12, no. 21: 9162.
https://doi.org/10.3390/su12219162