A robust meta-game for climate negotiations

Abstract

This paper deals with an application of the robust equilibrium concept in game theory to the assessment of the possible international agreement on climate that could be achieved in the conference of the parties negotiations organized by the UNFCCC. It is shown in particular that an acceptable, self-enforcing agreement could be obtained to maintain the temperature rise below 2$\,^\circ$C at the end of twenty-first century, with a balanced welfare loss among 11 groups of countries representing the parties limited to 1.8 % of their total discounted household consumption. To design this possible agreement we use a reduced order meta game where the players are the 11 groups of countries considered as the parties in negotiation, the strategies are the supply of emission quotas on an international emissions trading system and the payoffs are the net gains obtained from the emissions, trading and changes in the terms of trade minus the damage cost associated with the cumulative emissions during the 2010–2050 period. To identify the abatement costs that serve in the calculation of the payoffs and the gains due changes of terms of trade we use a statistical emulation of the GEMINI-E3 macroeconomic model. To obtain surrogate damage cost functions we introduce a coupled constraint in the game, imposing a limit to the cumulative emissions of all parties, which we call the global safety emissions budget. The multipliers intervening in the equilibrium necessary conditions are then interpreted as marginal damage costs. Games with coupled constraints admit a manifold of normalized equilibria and we show that they correspond to equilibria in games where each player is constrained by a given share of the safety emissions budget. Among all the normalized equilibria we look for the one which minimizes the maximum welfare loss, expressed in percentage of household consumption, among the 11 groups of countries. To take into account the uncertainty created by the statistical emulation approach and the approximate description of the emissions trading system we introduce robustness in the equilibrium computation.

Appendix A: Primer on robust game design

1.1 A.1 From Nash equilibrium to robust Nash equilibrium

We first recall the definition of a Nash equilibrium in a game in normal form. The game is defined by the given of:

• a set of players $j=\{1,2,\ldots ,m\}$;

• a set of strategies $S_j$ for each player $j\in M$, where $S_j\subset \mathbb {R}^{m_j}$ is convex;

• a set of payoffs $\phi _j(\mathbf {x}):\mathbf {x}\in \mathbf {S}=\times _{i=1}^m S_i\rightarrow \mathbb {R}$, where we assume that $\phi _j(\cdot )$ is continuously differentiable and concave in $x_j\in S_j$ and continuous in $\mathbf {x}^{-j}=(x_i)_{i\ne j}$.

Definition 1

A Nash equilibrium is a strategy $m$-tuple $\mathbf {x}^{\star } \in \mathbf {S}$ such that, for each player $j=1,\ldots ,m$, the following holds

$$\begin{aligned} \phi _j(\mathbf {x}^{\star })=\max _{x_j\in S_j} \phi _j([\mathbf {x}^{\star -j},x_j]), \end{aligned}$$

(33)

where $[\mathbf {x}^{\star -j},x_j])$ is the strategy $m$-tuple obtained by replacing in $\mathbf {x}^{\star }$ the component $x_j^\star$ by $x_j$.

If the strategy set of Player $j$ is defined by a set of inequality constraints

$$\begin{aligned} S_j=\left\{ x_j\in \mathbb {R}^{m_j} : h_j(x_j)\ge 0\right\} , \end{aligned}$$

(34)

where $h_j(\cdot ): \mathbb {R}^{m_j}\rightarrow \mathbb {R}^{p_j}$ is continuously differentiable and concave, and under usual constraint qualification conditions, a Nash equilibrium is characterized by the following first order conditions

$$\begin{aligned} \nabla _{x_j} \phi _j(\mathbf {x}) - \nu _j^T\frac{\partial }{\partial x_j} h_j(x_j)&= 0 \\ h_j(x_j)&\ge 0 \\ \nu _j&\ge 0 \\ \nu _j^T h_j(x_j)&= 0\\ j=1,\ldots ,m&\end{aligned}$$

Now we suppose that the payoff to Player $j$ is rewritten $\varphi _j(\mathbf {x},\xi )$, where $\varphi _j$ is differentiable in $x_j$ and continuous in $\mathbf {x}$ and $\xi _j$. Here $\xi _j$ is an uncertain parameter which takes value in an uncertainty set $\Xi _j$ which is supposed to be closed and bounded. We define the robust payoff for Player $j$ as the worst case function

$$\begin{aligned} \phi ^\rho (\mathbf {x})=\min _{\xi _j\in \Xi _j}\varphi _j(\mathbf {x},\xi _j). \end{aligned}$$

(35)

Definition 2

A robust equilibrium for the game defined by the uncertain payoffs $\varphi _j(\mathbf {x},\xi _j)$, is a Nash equilibrium for the game defined by the worst-case payoffs $\phi ^\rho (\mathbf {x})$.

1.2 A.2 From normalized equilibrium to robust normalized equilibrium

We now recall the definition of a normalized equilibrium in a game with coupled constraints. Assume, for the sake of simplicity, that in the game defined above, the strategy sets are $S_j=\mathbb {R}^{m_j}$. Let a coupled constraint set $\mathcal{{G}}\subset \times _{i=1}^m S_i$ be defined by a set of inequalities

$$\begin{aligned} g(\mathbf {x})\ge 0, \end{aligned}$$

(36)

where $g(\cdot ): \times _{i=1}^m S_i \rightarrow \mathbb {R}^p$ is continuously differentiable and concave.

Definiton 3

An equilibrium under the coupled constraint (36) is a strategy $m$-tuple $\mathbf {x}^{\star } \in \mathbf {S}$ such that, for each player $j=1,\ldots ,m$, the following holds

$$\begin{aligned} \phi _j(\mathbf {x}^{\star })=\max _{x_j\in S_j: [\mathbf {x}^{\star -j},x_j]\in \mathcal{{G}}} \phi _j([\mathbf {x}^{\star -j},x_j]). \end{aligned}$$

(37)

The equilibrium is normalized if there exists a multiplier $\mu$ and a vector of weights $\mathbf {r}=(r_j)_{j=1\ldots ,m}$ satisfying

$$\begin{aligned} r_j>0, \quad \sum _{j=1}^mr_j =1, \end{aligned}$$

such that the following first order conditions hold

$$\begin{aligned} \nabla _{x_j} \phi _j(\mathbf {x}) - \frac{\mu }{r_j}^T\frac{\partial }{\partial x_j} g(\mathbf {x})&= 0 \\ g(\mathbf {x})&\ge 0 \\ \mu&\ge 0 \\ \mu ^T g(\mathbf {x})&= 0\\ j=1,\ldots ,m.&\end{aligned}$$

We recall that there may exist a manifold of normalized equilibria, associated with the different possible weights $\mathbf {r}$.

Now we suppose that an uncertain parameter $\xi$ enters in the definition of the constraint, which writes now

$$\begin{aligned} g(x,\xi )\ge 0, \end{aligned}$$

(38)

where the parameter $\xi$ is element of an uncertainty set $\Xi$.

We define the robust coupled constraint set $\mathcal{{G}}^{\rho }\subset \times _{i=1}^m S_i$ defined by the inequalities

$$\begin{aligned} g^\rho (\mathbf {x})\ge 0, \end{aligned}$$

(39)

where for each component $k=1, \ldots ,p$

$$\begin{aligned} g_k^\rho (\mathbf {x})= \inf _{\xi \in \Xi }g_k(x,\xi ). \end{aligned}$$

(40)

Definition 4

A robust equilibrium under the uncertain coupled constraint (38) is a strategy $m$-tuple $\mathbf {x}^{\star } \in \mathbf {S}$ such that, for each player $j=1,\ldots ,m$, the following holds

$$\begin{aligned} \phi _j(\mathbf {x}^{\star })=\max _{x_j\in S_j: [\mathbf {x}^{\star -j},x_j]\in {\mathcal{{G}}}^{\rho }} \phi _j([\mathbf {x}^{\star -j},x_j]). \end{aligned}$$

(41)

1.3 A.3 The robust optimal game design problem

Consider a game with coupled constraint and its manifold of normalized equilibria, as defined in Definition 3. Let $\mathbf {x}^\star (\mathbf {r})$ be the equilibrium corresponding to the weight vector $\mathbf {r}=(r_j)_{j=1\ldots ,m},\,r_j>0,\,\sum _{j=1}^mr_j =1$.

Assume that there exists an upper-level decision maker who can select the weight vector $\mathbf {r}$ and who values the equilibrium outcome according to a function

$$\begin{aligned} \Psi :(\mathbf {x},j)\mapsto \mathbb {R}. \end{aligned}$$

(42)

The optimal game design problem consists then in finding the weight vector $\mathbf {r}\,^\circ$ which solves

$$\begin{aligned} \max _{\mathbf {r}} \min _j \Psi :(\mathbf {x}^\star (\mathbf {r}),j). \end{aligned}$$

(43)

The robust optimal game design problem consists then in finding the weight vector $\mathbf {r}\,^\circ$ which solves (43) where the equilibrium concept is replaced by the robust equilibrium concept defined above.