Bargaining over a climate deal: deadline and delay

Abstract

Assuming that a North-South transfer is the key to effective climate cooperation, we ask when and how much the North should offer to the South in return for a commitment to reduce deforestation and forest degradation. In light of the risk of irreversible damage over time, we examine a negotiation with a deadline. In this case, the North threatens the South over a negotiation dead-end in case an agreement is not reached rapidly. We assess the conditions for an agreement to be immediate or delayed, and discuss those situations likely to result in negotiation failure. Despite the risk of irreversible damage over time, we show that cooperation is likely to be delayed and we identify situations wherein the North and South do not reach an agreement within the deadline. Although Pareto-improving, cooperation may collapse because of inefficiencies related to incomplete information. What’s more, we show that in negotiations with a deadline, uncertainty about the benefits deriving from cooperation and the irreversibility of the damage that will be caused if cooperation is delayed are the two key components affecting choice.

Appendix

Proof of Propositions 2 and 3

Because they are related, we present a common proof for Propositions 2 and 3. This proof proceeds in two steps.

Step 1. We start by characterizing the best response strategies and study the three cases: when τ ₁ < $\widehat{\tau }$, when $\widehat{\tau }\leq \tau _{1}$ < c ⁺ and when τ ₁=c ⁺.

Case 1. If τ ₁=c ⁺, any type of South accepts the offer in the first period, an agreement is immediately reached, and the North obtains B−c ⁺.

Case 2. If $\widehat{\tau }\leq \tau _{1}$ < c ⁺, a coalition S ⁻ always accepts the offer in the first period, μ ₁(a∣c ⁻,τ ₁)=1, and a coalition S ⁺ always rejects it, μ ₁(a∣c ⁺,τ ₁)=0. In case that the South rejects the offer in the first period, the North knows with certainty that the South is a coalition S ⁺ and therefore offers a transfer c ⁺+α _S in the second period, which is always accepted. The payoff for the North then is $p_{1}^{-}(B-\tau _{1})+p_{1}^{+}\delta (B-\alpha _{N}-c^{+}-\alpha _{S})$ and is maximum when $\tau _{1}=\widehat{\tau }$.

Case 3. If c ⁻≤τ ₁ < $\widehat{\tau }$, coalition S ⁺ always rejects the offer in the first period, μ ₁(a∣c ⁺,τ ₁)=0. Let χ be the value of μ ₁(a∣c ⁻,τ ₁) such that the North is indifferent between offering c ⁺+α _S and c ⁻+α _S in the second period when μ ₁(a∣c ⁺,τ ₁)=0.

The North updates its beliefs in the second period over the type of S. We have:

For the North to be indifferent between offers c ⁺+α _S and c ⁻+α _S in the second period, χ should be such that $\frac{(1-\chi )p_{1}^{-}}{1-\chi p_{1}^{-}}(B-\alpha _{N}-c^{+}-\alpha _{S})+\frac{p_{1}^{+}}{1-\chi p_{1}^{-}}(B-\alpha _{N}-c^{+}-\alpha _{S})=\frac{(1-\chi )p_{1}^{-}}{1-\chi p_{1}^{-}}(B-\alpha _{N}-c^{-}-\alpha _{S})$ and therefore we have:

$$\chi =1-\frac{p_{1}^{+}(B-\alpha _{N}-c^{+}-\alpha _{S})}{p_{1}^{-}(c^{+}-c^{-})}$$

We can prove now that in this case, a S ⁻ coalition always adopts the mixed strategy χ in the first period.

First, note that if the North offers τ ₂(τ ₁)=c ⁺+α _S , its expected payoff in the second period is $p_{2}^{-}(\tau _{1})(B-\alpha _{N}-c^{+}-\alpha _{S})+p_{2}^{+}(\tau _{1})(B-\alpha _{N}-c^{+}-\alpha _{S})=B-\alpha _{N}-c^{+}-\alpha _{S}$. If it offers τ ₂(τ ₁)=c ⁻+α _S , its expected payoff is $p_{2}^{-}(\tau _{1})(B-\alpha _{N}-c^{-}-\alpha _{S})=(1-p_{2}^{+}(\tau _{1}))(B-\alpha _{N}-c^{-}-\alpha _{S})$.

We can deduce that the North adopts c ⁺+α _S with probability 1 in the second period only if $B-\alpha _{N}-c^{+}-\alpha _{S}>p_{2}^{-}(\tau _{1})(B-\alpha _{N}-c^{-}-\alpha _{S})$ and therefore if μ ₁(a∣c ⁻,τ ₁)>χ. The North adopts c ⁻+α _S with probability 1 in the second period only if μ ₁(a∣c ⁻,τ ₁)<χ and adopts a mixed strategy which we denote as [1−σ ₂(τ ₁),σ ₂(τ ₁)] offering c ⁺+α _S with probability σ ₂(τ ₁) if μ ₁(a∣c ⁻,τ ₁)=χ.

Second, note that if the North offers c ⁺+α _S in the second period, a coalition S ⁻ anticipates it and rejects any first period offer strictly lower than $\widehat{\tau }$. It follows that $\tau _{1}\in [ c^{-},\widehat{\tau }[$, μ ₁(a∣c ⁻,τ ₁)=0, which contradicts the previous statement. Similarly, if coalition S ⁻ anticipates that the North will offer c ⁻+α _S in the second period, it should accept any offers $\tau _{1}\in [ c^{-},\widehat{\tau }[$ in the first period, which also contradicts the first statement. We can deduce that the only feasible alternative is that a coalition S ⁻ always adopts a mixed strategy χ in the first period which fulfils our claim.

Given that a S ⁻ coalition plays a mixed strategy χ∈]0,1[ in the first period, χ is such that the expected payoff from accepting or rejecting offer τ ₁ is the same. We then have: τ ₁−c ⁻=δ(1−σ ₂(τ ₁))(c ⁻+α _S −c ⁻−α _S )+δσ ₂(τ ₁)(c ⁺+α _S −c ⁻−α _S ) and, therefore:

$$\sigma _{2}(\tau _{1})=\frac{\tau _{1}-c^{-}}{\delta (c^{+}-c^{-})}.$$

The expected payoff to the North associated with χ is $p_{1}^{-}\chi (B-\tau _{1})+p_{1}^{-}\frac{(1-\chi )p_{1}^{-}}{1-\chi p_{1}^{-}}\delta (B-\alpha _{N}-c^{-}-\alpha _{S})$ which is at a maximum when τ ₁=c ⁻. We can deduce that given that the North offers c ⁻ in the first period, it never offers c ⁺+α _S in the second period because σ ₂(τ ₁)=0. The offer in the second period is τ ₂(τ ₁)=c ⁻+α _S . At equilibrium, the North offers τ ₁=c ⁻ and τ ₂=c ⁻+α _S . A coalition S ⁻ accepts the offer with probability χ in the first period and always accepts the offer in the second, while a coalition S ⁺ always rejects the offer. The expected payoff for the North is then $p_{1}^{-}\chi (B-c^{-})+p_{1}^{-}\frac{(1-\chi )p_{1}^{-}}{1-\chi p_{1}^{-}}\delta (B-\alpha _{N}-c^{-}-\alpha _{S})$.

Step 2. Next we study the first period offer, that is the conditions for the North to offer $c^{-},\widehat{\tau }$ or c ⁺. We can deduce from proposition 1 that the North prefers to offer $\widehat{\tau }$ rather than c ⁺ as soon as c ⁺> $\overline{c^{+}}$. We start by proving that if W<c ⁺< $\overline{c^{+}}$, the North offers $\widehat{\tau }$ or c ⁻ rather than c ⁺. Consider the mapping g:ℝ→ℝ with

$$g(\delta )=\Pi _{H}-\Pi _{L}=B-c^{+}-p_{1}^{-}\chi (B-c^{-})-p_{1}^{-}\frac{(1-\chi )p_{1}^{-}}{1-\chi p_{1}^{-}}\delta (B-\alpha _{N}-c^{-}-\alpha _{S})$$

and where Π _H and Π _L respectively denote the North’s payoffs when the first period offer is high or low. Note that g(δ) is decreasing in δ and g(δ)=0 when $\delta =\overline{\delta }=\frac{(1-\chi p_{1}^{-})[c^{+}-\chi p_{1}^{-}c^{-}-(1-\chi p_{1}^{-})B]}{(p_{1}^{-})^{2}(\chi -1)(B-\alpha _{N}-c^{-}-\alpha _{S})}$. Given $\chi =1-\frac{p_{1}^{+}(B-\alpha _{N}-c^{+}-\alpha _{S})}{p_{1}^{-}(c^{+}-c^{-})}$, note that $\overline{\delta }\leq 0$ if and only if $c^{+}-\chi p_{1}^{-}c^{-}-(1-\chi p_{1}^{-})B\geq 0$, which is always true. We can deduce that $\overline{\delta }$ is always negative, and it follows that for any admissible δ∈[0,1], g(δ)<0 and the North always prefers the strategy (c ⁻,c ⁻+α _S ) to strategy (c ⁺,c ⁺+α _S ). The North never makes a high offer at the first period.

Next, we show that the choice for the North to play either $\widehat{\tau }$ or c ⁻ relies on the degradation of benefits over time. Define the mapping Δ:ℝ→ℝ with Δ=Π _M −Π _L and notice that Δ=0 when $\delta =\widetilde{\delta }=\frac{p_{1}^{+}(B-c^{-})(B-\alpha _{N}-c^{+}-\alpha _{S})}{(c^{-}-c^{+})Z}$ with $Z=(2p_{1}^{+}-1)(B-\alpha _{N}-c^{+}-\alpha _{S})+(1-p_{1}^{+})(c^{-}-c^{+})$ and $\widetilde{\delta }\leq 1$. Define ε∈ℝ such that δ= $\widetilde{\delta }+\varepsilon$ and note that Δ=Zε. We now consider the two cases where $p_{1}^{+}\gtrless 1/2$.

First, consider the case where $p_{1}^{+}<1/2$ and note that in this case $\widetilde{\delta }>0$ because Z<0. We can deduce that Δ is positive when ε<0 and negative when ε>0. The median offer is chosen as soon as $\delta <\widetilde{\delta }$.

Second, consider the case where $p_{1}^{+}>1/2$, a priori Z can either be positive or negative. We can prove that given c ⁺>W. Note that given $p_{1}^{+}>1/2$, Z is decreasing in c ⁺ and Z=0 when $c^{+}=\widetilde{c}=$ $\frac{(2p_{1}^{+}-1)(B-\alpha _{N}-\alpha _{S})+(1-p_{1}^{+})c^{-}}{p_{1}^{+}}$. $\widetilde{c}<W$ as soon as $\alpha _{N}+\alpha _{S}>\frac{(1-p_{1}^{+})^{2}(B-c^{-})}{1-2p_{1}^{+}}$, which is always true since by assumption, α _N +α _S ≥0 and $p_{1}^{+}>1/2$. We can deduce that $c^{+}>\widetilde{c}$ and therefore Z<0. We conclude that Δ is also positive as soon as $\delta <\widetilde{\delta }$, which completes the proof. □