# A Climate Alliance through Transfer: Transfer Design in an Economic Conflict Model

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## Abstract

**:**

## 1. Introduction

- Prize in the conflict model: savings of necessary effort in achieving the shared goal.
- Conflict activities: any actions, usually costly, which are detrimental to the shared goal.
- Conflict investments: the monetary representation of the conflict activities.
- Winning probability: share of the prize the player gets.

## 2. The Conflict Model

## 3. Design of the Transfer System

- STAGE 1: PLAYER 1 announces and pre-commits to the transfer program depending on the conflict investment.
- STAGE 2: Both PLAYER 1 and 2 simultaneously commit to conflict investments.
- STAGE 3: Conflict results are distributed. Settlement of the transfer depending on the commitment in STAGE 1 takes place, considering conflict investment in STAGE 2.

**a.**Holding the Cournot equilibrium ${x}_{1}={x}_{1}^{c}$ and expecting it.

**b.**${x}_{1}\to 0$ and expectation of it. Finally,

**c.**${x}_{1}$ is chosen so that the marginal benefits of increasing the transfer and those of increasing ${x}_{1}$ are equal at the same marginal cost, so the best option is always exhausted, which is expected by the PLAYERS.

**a.**resembles a possible morning-after effect, without adjustment of expectations in the conflict investments. In real-world considerations, the adjustment of expectations needs time, meaning that relatively high conflict investments need to be expected at the introduction of a transfer system. The scenario of a peaceful idyll is investigated in

**b.**, and it is shown that this scenario is not optimal. Instead, scenario

**c.**presents the new equilibrium to be expected in the long term.

**a.****Adherence to the Cournot equilibrium**From the status quo of a game consisting of an isolated STAGE 2, the expectation ${x}_{1}={x}_{1}^{c}$ could set in for PLAYER 2 or persist until proven wrong. On the other hand, PLAYER 1 may also show a tendency to traditionally hold on to a clear reference point despite the new situation of a transfer possibility. Likewise, this case serves as an analytical reference point. Thus, no changes occur here, except for the introduction of the transfer.For the Cournot equilibrium, the conflict investments for PLAYER 2 are ${x}_{2}^{o}={x}_{2}^{R}={x}_{2}^{c}$, and the objective function ${u}_{i}^{c}$ results. Insertion in (9) yields$${u}_{1}=\left\{\begin{array}{l}{v}_{1}-{x}_{1}^{c}-{u}_{2}^{c}for{x}_{2}=0\\ {u}_{1}^{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}for{x}_{2}0.\end{array}\right.$$Substituting ${x}_{1}^{c}$ and ${u}_{2}^{c}$ gives the “indirect” objective function$${u}_{1}=\left\{\begin{array}{l}{v}_{1}-\frac{{v}_{1}^{2}{v}_{2}}{{\left({v}_{1}+{v}_{2}\right)}^{2}}-\frac{{v}_{2}^{3}}{{\left({v}_{1}+{v}_{2}\right)}^{2}}for{x}_{2}=0\\ \frac{{v}_{1}^{3}}{{\left({v}_{1}+{v}_{2}\right)}^{2}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}for{x}_{2}0.\end{array}\right.$$The consideration of the situation, with respect to the accepted transfer and the conflict equilibrium, is as follows$${v}_{1}-\frac{{v}_{1}^{2}{v}_{2}}{{\left({v}_{1}+{v}_{2}\right)}^{2}}-\frac{{v}_{2}^{3}}{{\left({v}_{1}+{v}_{2}\right)}^{2}}=\frac{{v}_{1}^{3}}{{\left({v}_{1}+{v}_{2}\right)}^{2}}.$$If the left side is larger (smaller) than the right side, the transfer is (not) worthwhile for PLAYER 1. Equation (12) determines the situation in which PLAYER 1 is indifferent about providing a transfer or not, and yields the ratio that marks the break-even point at$${v}_{2}=0.5\left({v}_{1}+\sqrt{5}{v}_{1}\right)\approx 1.618{v}_{1}$$The following three numerical examples (numerical examples are rounded to whole numbers) illustrate the result:- (a)
- ${v}_{1}=1000;{v}_{2}=1600.$ The ratio is slightly higher than for indifference, so a transfer is worthwhile. This means ${u}_{2}^{c}\left(237;379\right)=606=$ $t\left({x}_{2}=0\right)$ and ${u}_{1}^{c}\left(237;379\right)=148<$ ${u}_{1}^{t}\left(237;0\right)=157$, while the target value of PLAYER 1 is ${u}_{1}^{t}={v}_{1}-{x}_{1}^{c}-{u}_{2}^{c}$ if the transfer is accepted. Thus, the societal gain from the transfer is ${u}_{1}^{t}-{u}_{1}^{c}=9$.The conflict investment ${x}_{2}^{c}=379$ can be avoided by the transfer. Shares in the prize towards the PLAYER with the lower valuation has a socially undesirable effect in the amount of $({v}_{1}-{v}_{2}){p}_{2}^{c}=-369$, depending on the share of the prize of PLAYER 2 in the Cournot equilibrium ${p}_{2}^{c}$.
- (b)
- ${v}_{1}=1000;{v}_{2}=1700.$ The ratio is smaller than for indifference, so a transfer is not worthwhile. This means ${u}_{2}^{c}\left(233;396\right)=674=$ $t\left({x}_{2}=0\right)$ and ${u}_{1}^{c}\left(233;396\right)=137>$ ${u}_{1}^{t}\left(237;0\right)=93$. Thus, the societal gain from the transfer is ${u}_{1}^{t}-{u}_{1}^{c}=-44$.The saving of ${x}_{2}^{c}=396$ cannot outweigh the damage caused by the redistribution of the prize towards the PLAYER with lower valuation amounting to $({v}_{1}-{v}_{2}){p}_{2}^{c}=-441$. In an investigation of the pre-play stage, this would be an argument to let PLAYER 2 be the transfer payer.
- (c)
- ${v}_{1}=1600;{v}_{2}=1000.$ The ratio is significantly higher than with indifference, so a transfer is worthwhile. This means ${u}_{2}^{c}\left(379;237\right)=148=$ $t\left({x}_{2}=0\right)$ and ${u}_{1}^{c}\left(379;237\right)=606<$ ${u}_{1}^{t}\left(237;0\right)=1073$. Thus, the societal effect of the transfer is ${u}_{1}^{t}-{u}_{1}^{c}=467$.Both the saving of ${x}_{2}^{c}=237$ and the redistribution of the prize to the PLAYER with higher valuation to the amount of $({v}_{1}-{v}_{2}){p}_{2}^{c}=231$ put PLAYER 1 in a better position.

**b.****Total peace**Since ${x}_{1}$ only enters ${u}_{2}$ as an expectation in the case of a transfer payment, the only costs that arise for ${x}_{2}=0$ are due to the then unnecessary ${x}_{1}$. This makes the corner solution ${x}_{1}\to 0$ appear to be gainful in saving these costs. However, this is only profitable if it is possible not to meet PLAYER 2′s expectations (i.e., if disappointment for PLAYER 2 is possible). If PLAYER 2 expects intended ${x}_{1}=\u03f5$, with $\u03f5$ as a very small number, the transfer amount $t\left({x}_{2}=0\right)={u}_{2}\left(\u03f5,{x}_{2}^{o}\right)={p}_{2}\left(\u03f5,{x}_{2}^{o}\right){v}_{2}-{x}_{2}^{o}$ must be much higher, since ${p}_{2}\left(\u03f5,{x}_{2}^{o}\right)$=${x}_{2}^{o}/\left(\u03f5+{x}_{2}^{o}\right)\to 1$ [54] (p. 106). In addition, any other small conflict investment by PLAYER 2 immediately leads to a large share of the prize.A numerical example illustrates this. Take example c) from above with $\u03f5=1$. Thus ${x}_{2}^{o}={x}_{2}^{R}\left(1\right)=31$ and ${u}_{2}\left(1;31\right)=938=$ $t\left({x}_{2}=0\right)>{u}_{2}^{c}\left(379;237\right)=148$. While the advantage of redistributing the prize is even more powerful, since ${p}_{2}\left(1;31\right)=0.969>{p}_{2}^{c}\left(379;237\right)=0.385$, the increase in the necessary transfer amount has a noticeable effect, as ${u}_{1}^{t}\left(1;0\right)=661<{u}_{1}^{t}\left(237;0\right)=1073$ illustrates.**c.****Marginal benefit calculus**As shown, because of the truthful expectations, ${x}_{1}=\u03f5$ is a strategically poor choice. However, the question of optimal ${x}_{1}$ remains. How can costs associated with conflict investment be saved without having to increase the transfer amount too much?PLAYER 1 can determine the optimal transfer amount by comparing the marginal benefits of the transfer for PLAYER 2 in the case of acceptance, with the amount of the marginal costs of further conflict investments for PLAYER 2; that is, the opportunity costs of accepting the transfer. Equating the two effects yields ${x}_{1}^{{t}^{*}}$ which maximizes the objective function of an optimal transfer ${t}^{*}$, which is expected to be accepted, indicated by the superscript index $*$.$$\frac{\partial {u}_{2}\left({x}_{2}=0\right)}{\partial t}=\left|\frac{\partial {u}_{2}({x}_{2}>0)}{\partial {x}_{1}}\right|,$$$$1=\left|-\frac{{x}_{2}{v}_{2}}{{\left({x}_{1}+{x}_{2}\right)}^{2}}\right|.$$Rearranging results in$${x}_{1}^{{t}^{*}}\left({x}_{2}\right)=\sqrt{{x}_{2}{v}_{2}}-{x}_{2}.$$The marginal costs of transfer and conflict are assumed to be constant and equal for PLAYER 1, which is why they do not need to be considered further in the calculation. The marginal net gains of the two alternatives must be compared, with both sides reduced by the same cost.$$\frac{\partial {u}_{1}\left({x}_{2}=0\right)}{\partial t}=-1=\frac{\partial {u}_{1}}{\partial {x}_{1}}.$$This situation results in the following solution by substituting the reaction function of PLAYER 2, which equals the expectation of PLAYER 1 regarding PLAYER 2′s expectation of PLAYER 1′s action:$${x}_{1}^{{t}^{*}}\left({x}_{2}^{R}\right)=0,25{v}_{2}.$$The reaction of PLAYER 2 given by formula (3) is also$${x}_{2}^{R}\left({x}_{1}^{{t}^{*}}\right)=0,25{v}_{2}.$$For ${x}_{1}>{x}_{1}^{{t}^{*}}$, an increase in the attractiveness of the transfer option would be worthwhile to reduce ${x}_{1}$ further, while for ${x}_{1}<{x}_{1}^{{t}^{*}}$, an increase in ${x}_{1}$ can result in a worthwhile reduction in the transfer.The respective optimal transfer is$${t}^{*}\left({x}_{1}^{{t}^{*}},{x}_{2}^{R}\right)=\left\{\begin{array}{c}{u}_{2}\left({x}_{1}^{{t}^{*}},{x}_{2}^{o}\right)for{x}_{2}=0\\ 0\hspace{1em}\hspace{1em}else,\end{array}\right.$$$${t}^{*}\left({v}_{2}\right)=\left\{\begin{array}{c}{u}_{2}\left(0,25{v}_{2},0,25{v}_{2}\right)for{x}_{2}=0\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}else.\end{array}\right.$$It is worth noting that ${x}_{1}^{{t}^{*}}$ does not depend on ${v}_{1}$. This is because by setting ${x}_{1}$ independent of ${v}_{1}$, it is not included in the calculus of PLAYER 2 (this circumstance could change if PLAYER 1 experiences liquidity problems). In general, this gives ${x}_{2}^{o}=0,25{v}_{2}$, ${t}^{*}={u}_{2}=0,25{v}_{2}$ and ${u}_{1}^{{t}^{*}}={v}_{1}-0,5{v}_{2}$.For the numerical example c), ${x}_{1}^{{t}^{*}}=250=$ ${x}_{2}^{R}\left(250\right)$. It is ${u}_{2}\left(250;250\right)=250=$ ${t}^{*}\left({x}_{2}=0\right),$ and ${u}_{1}^{c}\left(379;237\right)=606<$ ${u}_{1}^{t}\left(237;0\right)=1073<{u}_{1}^{{t}^{*}}\left(250;0\right)=1100$. Thus, the social effect due to the optimal transfer is ${u}_{1}^{{t}^{*}}-{u}_{1}^{c}=494$. In addition to the saving of ${x}_{2}^{c}=237$ and the redistribution of the prize to the PLAYER with higher valuation (amounting to $({v}_{1}-{v}_{2}){p}_{2}^{c}=231$), the saving of ${x}_{i}^{c}-{x}_{1}^{{t}^{*}}=129$ is now added, reduced by the increase in the transfer ${t}^{*}\left({x}_{2}=0\right)-t\left({x}_{2}=0\right)=102$.The deviation of ${x}_{1}^{{t}^{*}}$ to a higher (lower) value of ${x}_{1}=300\left(200\right)$, which would equally be anticipated by PLAYER 2, could only result in the target value ${u}_{1}^{t}=1095\left(1094\right)$. Remarkable is the accompanying change of ${u}_{2}^{t}=t=205\left(306\right)$, which, however, does not enter the calculation of PLAYER 1. Thus, the donor has a first mover advantage.

## 4. Peace as a Public Good

## 5. Conclusions

_{2}savings, coal phase-out, and the establishment of nature reserves. The model suggests incentivizing through a transfer. For the case of international climate agreements, any international agent could start to declare their respective intention. For a large climate agreement and in order for such a transfer to be credibly offered, it is necessary to form and equip a corresponding organization for climate protection, such as the WTO for trade, which must be taken into account with the design of a global environmental governance [68,69] and currently represents a functioning model for international cooperation [70]. The conference of the parties may be the appropriate institution for this task.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$1$ | Index for PLAYER 1, which is the transfer payer |

$2$ | Index for PLAYER 2, which is a transfer recipient |

$\u03f5$ | A very small number |

$*$ | Superscript index for the transfer that maximizes the objective function |

$c$ | Superscript index for Cournot equilibrium |

CSF | Contest success function |

$h$ | Other members of society than PLAYER 1 and PLAYER 2 |

$i$ | Index for the $\mathrm{PLAYERs},i\in \left\{1,2\right\}$ |

$j$ | Index for the respective competitor, i.e., the other PLAYER |

$k$ | Other members of society than PLAYER 1 |

$M$ | Set of several actors in a society |

$o$ | Superscript index for action rejecting the transfer |

${p}_{i}$ | Probability of winning, mathematical representation of the contest success function (CSF) |

${p}_{2}^{c}$ | Probability of winning of PLAYER 2 in the Cournot equilibrium |

$R$ | Superscript index for reaction function $t$ Transfer program, also may be used as superscript index for actions in accordance with the incentives by the transfer scheme |

${t}^{*}$ | Optimal transfer program |

${t}_{k}$ | Transfer program targeting all other PLAYERs |

${u}_{i}$ | Target function, also known as objective function |

${u}_{1}^{t}$ | Target value for PLAYER 1 if the transfer is accepted by PLAYER 2 |

${u}_{i}^{c}$ | Target function of the Cournot equilibrium |

${v}_{i}$ | Valuation of the total potential prize |

${x}_{i}$ | Conflict efforts |

${x}_{i}^{c}$ | Conflict efforts in the Cournot equilibrium |

${x}_{i}^{R}$ | Reaction function |

${x}_{1}^{{t}^{*}}$ | Conflict effort of PLAYER 1 that maximizes the utility of an expectedly accepted transfer |

${x}_{2}^{o}$ | Conflict efforts by PLAYER 2 if the transfer is rejected |

${x}_{2}^{t}$ | Conflict efforts by PLAYER 2 if the transfer is accepted |

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## Share and Cite

**MDPI and ACS Style**

Franke, M.; Neumärker, B.K.J.
A Climate Alliance through Transfer: Transfer Design in an Economic Conflict Model. *World* **2022**, *3*, 112-125.
https://doi.org/10.3390/world3010006

**AMA Style**

Franke M, Neumärker BKJ.
A Climate Alliance through Transfer: Transfer Design in an Economic Conflict Model. *World*. 2022; 3(1):112-125.
https://doi.org/10.3390/world3010006

**Chicago/Turabian Style**

Franke, Marcel, and Bernhard K. J. Neumärker.
2022. "A Climate Alliance through Transfer: Transfer Design in an Economic Conflict Model" *World* 3, no. 1: 112-125.
https://doi.org/10.3390/world3010006