A coordination game model for risk allocation of a PPP project with the weakened hedged probabilistic linguistic term information

Figures & data

Figure 1. The number of novel projects for PPP library at the end of September.

Source: calculated from authors.

Table 1. The comparison of different techniques.

The techniques	Syntax	Mathematical formulation
HFLTSs	greater than…; lower than…; between… and…	$\left\{s_{\alpha -1},s_{\alpha },s_{\alpha +1}\right\}$
PLTSs	atomic + possible degree	$\left\{{(s_{\alpha },p)}^{(k)}\right\}$
LTWHs	weakened hedge + atomic term	$〈h_t,s_{\alpha }〉$
P-LTWHs	weakened hedge + atomic + possible degree	$({〈h_t,s_{\alpha }〉}^{(k)},p^{(k)})$

Source: calculated from authors.

Table 2. The payoff matrix with preference view.

		Private
		Allocation	Non-allocation
Public	Allocation	$F-C+\delta$	$\delta$
	Non-allocation	$0$	$F^{\prime }$

* For the preference view matrix, we take partners have preference to ‘allocation’ as an example, the preference to other strategies are similar.

Source: calculated from authors.

Table 3. The payoff matrix with equivocators.

		Private
		Allocation	Non-allocation	Hesitant
Public	Allocation	$F-C$	0	r
	Non-allocation	0	$\begin{array}{c}{F}^{\prime }\hfill \end{array}$	$F^{\prime }-C-r$
	Hesitant	r	$F^{\prime }-C-r$	$F^{\prime }$

Source: calculated from authors.

Table 4. The risk allocation of CRFs.

	CRFs	Allocation stakeholders
Feasibility stage	Risk of approval delay	Public sector
	Design change risk	Public sector
	Policy risk	Public sector
	Stability risk	Project company
Construction stage	Investment control risk	Public and Private
	Project schedule risk	Construction sector
	Project quality risk	Construction sector
	Project cost risk	Construction sector
Operation stage	Market risk	Public and Private

Source: calculated from authors.

Figure 2. The partners structure of the Lintong line.

Source: calculated from authors.

Table 5. The benefits of different partners in three aspects.

	Public	Private
The environment	$\left\{(〈h_0,s_4〉,0.75),(〈h_0,s_3〉,0.25)\right\}$	$\left\{(〈h_1,s_6〉,0.5),(〈h_2,s_5〉,0.5)\right\}$
The economic	$\left\{(〈h_0,s_5〉,0.8),(〈h_0,s_2〉,0.2)\right\}$	$\left\{(〈h_0,s_1〉,0.5),(〈h_1,s_6〉,0.5)\right\}$
The social	$\left\{(〈h_0,s_3〉,0.4),(〈h_1,s_6〉,0.6)\right\}$	$\left\{(〈h_0,s_5〉,0.65),(〈h_2,s_6〉,0.35)\right\}$

Source: calculated from authors.

Table 6. The cost of partners in three aspects.

	Public	Private
Equity investment expenditure	$\left\{(〈h_0,s_4〉,0.6),(〈h_1,s_3〉,0.4)\right\}$	$\left\{(〈h_0,s_5〉,0.75),(〈h_2,s_2〉,0.25)\right\}$
Operating subsidy expenditure	$\left\{(〈h_0,s_6〉,0.5),(〈h_2,s_3〉,0.5)\right\}$	$\left\{(〈h_0,s_6〉,0.8),(〈h_1,s_5〉,0.2)\right\}$
Risk sharing expenditure	$\left\{(〈h_0,s_3〉,0.35),(〈h_2,s_4〉,0.65)\right\}$	$\left\{(〈h_1,s_4〉,0.75),(〈h_0,s_5〉,0.25)\right\}$

Source: calculated from authors.

Table 7. The data of different partners in allocating market risk.

	Public	Private
The benefit	$\left\{(〈h_0,s_4〉,0.65),(〈h_1,s_6〉,0.35)\right\}$	$\left\{(〈h_1,s_4〉,0.55),(〈h_2,s_6〉,0.45)\right\}$
The cost	$\left\{(〈h_0,s_4〉,0.48),(〈h_2,s_3〉,0.52)\right\}$	$\left\{(〈h_1,s_5〉,0.76),(〈h_2,s_4〉,0.24)\right\}$
other benefits	$\left\{(〈h_0,s_2〉,0.6),(〈h_0,s_1〉,0.4)\right\}$	$\left\{(〈h_1,s_0〉,0.57),(〈h_0,s_1〉,0.43)\right\}$

Source: calculated from authors.

Table 8. The payoff matrix with preference view.

		Private
		Allocate	Non-allocate
Public	Allocate	$1.19+\delta , 0.035+\delta$	$\delta , \delta$
	Non-allocate	$0, 0$	$1.6, 0.215$

Source: calculated from authors.

Table 9. The payoff matrix with equivocators.

		Private
		Allocate	Non-allocate	Hesitant
Public	Allocate	$1.19, 0.035$	$0, 0$	$r, r$
	Non-allocate	$0, 0$	$\begin{array}{c}1.6, 0.215\hfill \end{array}$	$0.44-r, -0.975-r$
	Hesitant	$r, r$	$0.44-r, -0.975-r$	$1.6, 0.215$

Source: calculated from authors.

Figure 3. The dynamic graph with preference view game model of the government.

Source: calculated from authors.

Figure 4. The dynamic graph with preference view game model of the private.

Source: calculated from authors.

Figure 5. The dynamic graph with equivocators game model of the government.

Source: calculated from authors.

Figure 6. The dynamic graph of the government with equivocators model.

Source: calculated from authors.

Table 10. The matrix with preference view and equivocators.

		Private
		Allocate	Non-allocate	Hesitant
Public	Allocate	$F-C+\delta$	δ	$r+\delta$
	Non-allocate	0	$\begin{array}{c}{F}^{\prime }\hfill \end{array}$	$F^{\prime }-C-r$
	Hesitant	r	$F^{\prime }-C-r$	$F^{\prime }$

Source: calculated from authors.

Table 11. The payoff matrix with preference view and equivocators.

		Private
		Allocate	Non-allocate	Hesitant
Public	Allocate	$1.19+\delta ,0.035+\delta$	δ	$r+\delta ,r+\delta$
	Non-allocate	0,0	$\begin{array}{c}1.6, 0.29\hfill \end{array}$	$0.44-r,-0.975-r$
	Hesitant	$r, r$	$0.44-r,-0.975-r$	$1.6, 0.215$

Source: calculated from authors.

Figure 7. The dynamic graph with preference view and equivocators game model of the government.

Source: calculated from authors.

Figure 8. The dynamic graph with preference view and equivocators game model of the private.

Source: calculated from authors.