### 1. Introduction

### 2. Methods

*t*= 0) the two coalitions negotiate to make concessions. In the second stage (

*t*= 1) concessions to reduce GHG emission are realized. Such approach converts the static concession game in to a dynamic one. At the end of the concession both coalitions are expected to reduce their status quo emission levels.

##### (1)

$${R}_{i}({e}_{i}^{1},{e}_{i}^{2n},\cdots ,{e}_{i}^{n},{R}_{j})={R}_{j}({e}_{j}^{1},{e}_{j}^{2},\cdots ,{e}_{j}^{n},{R}_{i})$$*R*

*(•) and*

_{i}*R*

*(•) are reduction functions for the two coalitions (*

_{j}*i*,

*j*), ${e}_{i}^{1},{e}_{i}^{2},\cdots ,{e}_{i}^{n}$ and ${e}_{j}^{1},{e}_{j}^{2},\cdots ,{e}_{j}^{n}$ are emission reduction decision variables. The coalitions’ emission reductions are dependent on their emission reduction decision variables and each other’s emission reductions.

*R*

*(•) /*

_{i}*R*

*(•) =*

_{j}*w*

*. Where*

_{j}*w*

*is the concession of coalition*

_{j}*j*relative to coalition

*i*. This approach helps to protect the interests of the negotiating coalitions and avoid the cost associated with taking too much time to reach a compromising decision on emission reduction. Now suppose that the two coalitions carried out concessions based on their negotiation strategy. ${s}_{i}^{1}\in [0,{x}_{i}^{0}]$ is the amount of emission reduction coalition

*i*made at

*t*= 1 based on its status quo emission variable ${x}_{i}^{0}$. The Nash equilibrium decision value for coalition

*i*at

*t*= 0 is ${x}_{i}^{0}$. The respective emission reduction for coalition

*j*is ${w}_{j}{x}_{i}^{0}$. If coalition

*i*reduce ${x}_{i}^{0}$ the same applies for coalition

*i*who is involved in the emission reduction game with coalition

*j*. Hence, the following equality is satisfied.

##### (2)

$${s}_{i}^{1}=\{\begin{array}{l}{x}_{i}^{0},{w}_{i}{s}_{j}^{1}\ge {x}_{i}^{0}\hfill \\ {w}_{i}{s}_{j}^{1},{w}_{i}{s}_{j}^{1}<{x}_{i}^{0}\hfill \end{array}$$*t*= 0. Eq. (2) also shows that each coalition has ${s}_{i}^{1}$ and

*w*

*as decision variables. Winner or loser in the negotiation process is determined by the bargaining power of the negotiating parties.*

_{i}*t*= 1 the rule to judge the winner and the loser in the negotiation process is marked out as follows;

##### (3)

$$\frac{{w}_{i}\raisebox{1ex}{${s}_{j}^{1}$}\!\left/ \!\raisebox{-1ex}{${x}_{i}^{0}$}\right.}{\raisebox{1ex}{${s}_{i}^{1}$}\!\left/ \!\raisebox{-1ex}{${x}_{j}^{0}$}\right.}>\frac{{w}_{j}\raisebox{1ex}{${s}_{i}^{1}$}\!\left/ \!\raisebox{-1ex}{${x}_{j}^{0}$}\right.}{\raisebox{1ex}{${s}_{i}^{1}$}\!\left/ \!\raisebox{-1ex}{${x}_{i}^{0}$}\right.}$$##### (4)

$${w}_{i}\frac{{x}_{j}^{0}}{{x}_{i}^{0}}>{w}_{j}\frac{{x}_{i}^{0}}{{x}_{j}^{0}}\Rightarrow {w}_{i}>{w}_{j}{\left(\frac{{x}_{i}^{0}}{{x}_{j}^{0}}\right)}^{2}$$*w*

*>*

_{i}*Mw*

*then coalition*

_{j}*i*is the winner at

*t*= 1 and bids a greater proportion of concession than the relative concession decision variables of the loser which is

*j*. On the other hand if

*w*

*>*

_{j}*Mw*

*then coalition*

_{i}*j*is the winner at

*t*= 1 and bids a greater proportion of concession than the relative concession decision variables of the loser which in this case is coalition

*i*. When both sides of Eq. (5) are equal random way is used to determine the winner and the loser [29].

*i*is the winner and

*w*

*is its corresponding offer, then as loser coalition*

_{i}*j*makes a concession first. Its concession which is the function of

*w*

*will be*

_{i}*s*

*(*

_{j}*w*

*). Therefore utility function of coalition*

_{i}*j*will be;

##### (6)

$${y}_{j}^{L}({w}_{i})={u}_{j}({x}_{i}^{0}-{w}_{i}{s}_{j}({w}_{i}),{x}_{j}^{0}-{s}_{j}({w}_{i}))$$**as winner is can be written as;**

*i*##### (7)

$${y}_{i}^{W}({w}_{i})={u}_{i}({x}_{i}^{0}-{w}_{i}{s}_{j}({w}_{i}),{x}_{j}^{0}-{s}_{j}({w}_{i}))$$### 3. Results and Discussion

##### (8)

$$\{\begin{array}{l}{y}_{i}^{0}={a}_{i}{x}_{i}^{0}-{b}_{i}{({x}_{i}^{0})}^{2}-{b}_{i}{x}_{i}^{0}{x}_{j}^{0}\hfill \\ {y}_{i}^{0}={a}_{j}{x}_{j}^{0}-{b}_{j}{*{s}_{j}^{0})}^{2}-{b}_{j}{x}_{i}^{0}{x}_{j}^{0}\hfill \end{array}$$*a*

*> 0 is the logistic coefficient showing the economic benefits. The emissions of one coalition have negative environmental and economic externalities on the other coalition. One’s utility function is a decreasing function of other’s GHG emission. Therefore*

_{i}*b*

*> 0 is the constraining parameter. According to the law of diminishing marginal utility the graph of utility function of each negotiating agent is concave.*

_{i}##### (9)

$$\{\begin{array}{l}\frac{d{y}_{i}^{0}}{d{x}_{i}^{0}}={a}_{i}-2{b}_{i}{x}_{i}^{0}-{b}_{i}{x}_{j}^{0}\hfill \\ \frac{d{y}_{j}^{0}}{d{x}_{j}^{0}}={a}_{j}-2{b}_{j}{x}_{j}^{0}-{b}_{j}{x}_{i}^{0}\hfill \end{array}$$##### (10)

$$\{\begin{array}{l}{x}_{i}^{0}=\frac{2{a}_{i}{b}_{j}-{a}_{j}{b}_{i}}{3{b}_{i}{b}_{j}}\hfill \\ {x}_{j}^{0}=\frac{2{a}_{j}{b}_{i}-{a}_{i}{b}_{j}}{3{b}_{i}{b}_{j}}\hfill \end{array}$$##### (11)

$$\begin{array}{c}2{a}_{i}{b}_{i}-{a}_{j}{b}_{i}\ge 0\hspace{0.17em}\text{and\hspace{0.17em}}2{a}_{j}{b}_{i}-{a}_{i}{b}_{j}-{a}_{i}{b}_{j}\ge 0\\ \{\begin{array}{l}{y}_{i}^{0}=\frac{{(2{a}_{i}{b}_{j}-{a}_{j}{b}_{i})}^{2}}{9{b}_{i}{({b}_{j})}^{2}}\hfill \\ {h}_{j}^{0}=\frac{{(2{a}_{j}{b}_{j}-{a}_{i}{b}_{j})}^{2}}{9{({b}_{i})}^{2}{b}_{j}}\hfill \end{array}\end{array}$$*i*is the winner during the negotiation process coalition

*j*as the loser cuts its status quo GHG emission first and its utility function will be;

##### (12)

$${y}_{i}^{1L}={a}_{j}({x}_{j}^{0}-{s}_{j})-{b}_{j}({x}_{j}^{0}-{s}_{j})-{b}_{j}({x}_{i}^{0}-{w}_{i}{s}_{j})({x}_{j}^{0}-{s}_{j})$$*j*

##### (13)

$${x}_{j}({w}_{i})={S}^{\prime}({w}_{i})=\frac{{w}_{i}(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}{6{b}_{i}{b}_{j}({w}_{i}+1)}$$##### (14)

$${x}_{j}^{1}={x}_{j}^{0}-{S}^{\prime}=\frac{({w}_{i}+2)(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}{6{b}_{i}{b}_{j}({w}_{i}+1)}$$*i*will be;

##### (15)

$${s}_{i}({s}_{j})={w}_{i}{S}^{\prime}=\frac{{({w}_{i})}^{2}(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}{6{b}_{i}{b}_{j}({w}_{i}+1)}$$##### (16)

$${x}_{i}^{1}={x}_{i}^{0}-{w}_{i}{S}^{\prime}=\frac{2(2{a}_{1}{b}_{2}-{a}_{2}{b}_{1})({w}_{i}+1)}{6{b}_{i}{b}_{j}({w}_{i}+1)}-\frac{{({w}_{i})}^{2}(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}{6{b}_{i}{b}_{j}({w}_{i}+1)}$$##### (17)

$$\{\begin{array}{l}{y}_{i}^{1W}=({a}_{i}-{b}_{i}({x}_{j}^{0}-{s}_{j}))({x}_{i}^{0}-{w}_{i}{s}_{j})-{b}_{i}{({x}_{i}^{0}-{w}_{i}{s}_{j})}^{2}\hfill \\ {y}_{j}^{2L}=n({a}_{j}-{b}_{j}({x}_{i}^{0}-{s}_{j})-{b}_{j}{({x}_{j}^{0}-{s}_{j})}^{2}\hfill \end{array}$$*j*is assumed to be the winner its emission reduction functions will be the following;

##### (18)

$$\{\begin{array}{l}{s}_{i}^{n}=\frac{{W}_{j}(2{a}_{i}{b}_{j}-{a}_{j}{b}_{i})}{6{b}_{i}{b}_{j}({w}_{i}+1)}\hfill \\ {s}_{j}^{n}({s}_{i}^{n})=\frac{{({w}_{j})}^{2}(2{a}_{i}{b}_{j}-{a}_{j}{b}_{i})}{2{b}_{i}{b}_{j}({w}_{j}+1)}\hfill \end{array}$$##### (19)

$$\{\begin{array}{l}{y}_{i}^{1L}=({a}_{i}-{b}_{i}({x}_{i}^{0}-{w}_{j}{s}_{i}^{nn})({x}_{i}^{0}-{s}_{i}^{n})-{b}_{i}{({x}_{i}^{0}-{s}_{i}^{n})}^{2}\hfill \\ {y}_{j}^{2W}=({a}_{j}-{b}_{j}({x}_{i}^{0}-{s}_{i}^{n}))({x}_{j}^{0}-{w}_{j}{s}_{i}^{n})-{b}_{j}{({x}_{j}^{0}-{w}_{j}{s}_{i}^{n})}^{2}\hfill \end{array}$$*i*’s bid

*w*

*have equal preference with coalition*

_{i}*j*’s bid ${w}_{j}=\frac{{w}_{i}}{M}$ during negotiations. Therefore, coalition

*i*choose the optimal strategy that ensures ${y}_{i}^{1W}({w}_{i})={y}_{i}^{1L}({w}_{i}/M)$ thus;

##### (20)

$$\{\begin{array}{l}{w}_{i}={W}_{i}({a}_{1},{a}_{2},{b}_{3},{b}_{4})>0\hfill \\ {w}_{j}={W}_{j}({a}_{i},{a}_{2},{b}_{3},{b}_{4})>0\hfill \end{array}$$*W*

*>*

_{i}*MW*

*coalition*

_{j}*i*will be the winner and coalition

*j*the loser otherwise

*i*will be loser and coalition

*j*will be the winner. When coalition

*i*is the winner and coalition

*j*the loser their respective GHG emission reductions will be

*w*

_{i}*S*′ and

*S*′ respectively. In this case their utility functions will be;

##### (21)

$$\{\begin{array}{l}{y}_{i}^{1W}=({a}_{i}-{b}_{i}({x}_{j}^{0}-{S}^{\prime}))({x}_{i}^{0}-{w}_{i}{S}^{\prime})-{b}_{i}{({x}_{i}^{0}-{w}_{i}{S}^{\prime})}^{2}\hfill \\ {y}_{j}^{2L}=({a}_{j}-{b}_{j}({x}_{i}^{0}-{w}_{i}{S}^{\prime}))({x}_{j}^{0}-{S}^{\prime})-{b}_{j}{({x}_{j}^{0}-S{1}^{\prime})}^{2}\hfill \end{array}$$##### (22)

$$\mathrm{\Delta}{y}_{i}^{1W}=\frac{({a}_{i}{b}_{j}({({w}_{i})}^{2}+4)(2{a}_{j}{n}_{i}-{a}_{i}1{b}_{j})}{36({w}_{i}+1){b}_{i}{({b}_{j})}^{2}}-\frac{2{a}_{j}{b}_{i}({({w}_{i})}^{2}+1)(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}{36({w}_{i}+1){b}_{i}{({b}_{j})}^{2}}$$##### (23)

$$\mathrm{\Delta}{y}_{j}^{2L}=\frac{{({w}_{i}+1)}^{2}{(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}^{2}}{36({w}_{i}+1){({b}_{i})}^{2}{b}_{j}}$$*i*depends on its logistic and constraint parameters as well as on the utility improvement of coalition

*j*. The converse is also true when coalition

*j*is a winner.

*i*as a winner of the negotiation will improve its utility. If ${W}_{i}<2\sqrt{{\scriptstyle \frac{2{a}_{i}{b}_{j}-{a}_{j}{b}_{i})}{2(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}}}$ the utility of coalition

*i*will decrease. Since the game is symmetric when coalition

*j*is winner even though coalition

*i*is loser its utility improvement is similar to its utility improvement it can secure as a winner. From the above analysis the following conclusions can be reached. First, the utilities of both the bargaining parties will not improve during the first stage of the negotiation process. During, the second phase of negations because both parties makes reductions in emissions their utilities improve. The utility of coalition

*j*improves since it is the loser in the negotiation process while the utility of coalition

*i*will improve as well as long as it is with in the balanced emission range. Generally winners have high GHG emission reduction ratios. Second, if the two coalitions act unilaterally to maximize their benefits they may not achieve pareto-optimality and end up with “tragedy of the commons” scenario. The negotiating coalitions cannot achieve pareto-optimality by making concessions as well, but if their GHG emission reductions are within a certain balanced range both the negotiating parties can achieve pareto-improvement.

### 4. Hypothetical Example

*i*and

*j*. The two coalitions emit GHG mainly due to consumption of fossil fuels. Emission of GHG has negative environmental consequences. The two coalitions acknowledge the problem and are willing to reduce GHG emissions through negotiations. Therefore concession game can be used to analyze their emission reduction plans and determine the balanced emission range that results in utility improvement for both negotiating coalitions. The utility functions of the bargaining coalitions can be written as the following;

*t*= 0 maximizing the above utility functions results the following values for the emissions and utilities of the negotiating coalitions.

*i*is the winner and coalition

*j*is loser. If coalition

*j*reduces emissions by a unit then coalition

*i*should reduce by

*w*

*units. Therefore, the utility function of coalition*

_{i}*j*will be;

*i*and

*j*.

*j*is winner and coalition

*i*is loser their emission reduction functions of the coalitions will be;

*i*’s bid

*w*

*and coalition*

_{i}*j*’s bid

*w*

*are the keys to determine the winner and the loser in the negotiation process. Weather the coalition is winner or a loser it gains the same utility improvement.*

_{j}*w*

*is equal to 1.3780 and When ${y}_{j}^{1W}({w}_{j})={y}_{2}^{1L}(2.0410{w}_{j})$*

_{i}*w*

*is equal to 0.2494. Therefore for the case*

_{j}*w*

*>2.0410*

_{i}*w*

*coalition*

_{j}*i*is the winner and coalition

*j*is the loser. Their respective reductions

*s*

*and*

_{j}*s*

*will be 0.1127 and 0.1533. Hence at the end of the concession the condition $M{W}_{j}<{W}_{i}<2\sqrt{{\scriptstyle \frac{(2{a}_{i}{b}_{j}-{a}_{j}{b}_{i})}{2(2{a}_{j}{b}_{i}-{a}_{i}{b}_{j})}}}$ is satisfied. This indicates that the utilities of the two bargaining coalitions are improved. The emission levels after reduction and utilities of the negotiating coalitions will be;*

_{i}