J. Laurent-Lucchetti (U. Bern) J. Leroux and B. Sinclair-Desgagné (HEC Montréal) Splitting an uncertain (natural) capital

Introduction Topic: Strategic behavior under the threat of dramatic events when experts disagree. Issues of climate change: o shutdown of thermohaline circulation, o permafrost meltdown… Other issues: o epidemiological outbreaks, o resistance to antibiotics…

Introduction (cont.) How are agents expected to behave? Can risk aversion alone help avoid dramatic outcomes? Is there a role for coordination devices (i.e., Kyoto protocol, COP15, etc.) ? What are the implications for precautionary policies?

Related Literature Decision theory, Nash demand game, commons problem. Bramoullé &Treich (JEEA, 2009): global public bad context. o Emissions are always lower under uncertainty and welfare may be higher. o Cooperation is less likely under uncertainty (gains from cooperation are lower). Nkuiya (2011), Morgan & Prieur (2011). Contribution game to discrete public good: Nitzan and Romano, JPubE, 1990; McBride, JPubE, 2006; Barbieri and Malueg, 2009; Rapoport (many); Dragicevic and Engle-Warnick, 2011.

Contribution Introduce a strong discontinuity in the (uncertain) available amount of a common resource. In sharp contrast with previous results, introducing uncertainty does not always lead to lower consumption, even if all agents are risk averse. “Dangerous" equilibria may exist, where agents behave as if ignoring the possibility of a bad outcome, even if all agents are risk averse.

Contribution (cont.) Cooperation can be beneficial to all agents. Under mild conditions, a move from an dangerous eq. to a “cautious” eq. is a Pareto improvement. Support for precautionary policies and coordination devices.

The Simple Model

n agents simultaneously consume a common resource. The amount of resource available, r, is uncertain: r = Each agent i chooses a consumption level: x i ≥ 0. If ∑x i ≤ r, each agent receives x i. Otherwise, they receive nothing. Simultaneous “Divide the dollar” game with uncertainty on “the dollar”. 1 with prob. p<1 a < 1 with prob. (1-p)

The Simple Model (cont.) Utility of an agent i: u i (x i ). u i ’s are concave (risk aversion and risk neutrality) non- decreasing and u i (0)=0. Expected payoff of an agent i: v i (x i, X -i )= u i (x i ) Ι(X≤a)+p u i (x i ) Ι(a<X≤1) where X = ∑x i = x i +X -i

Best responses For each agent i: If X -i > a: o If X -i < 1: demand x i = 1-X -i o If X -i ≥ 1 : demand x i = 0 or anything higher. If X -i ≤ a: o Demand x i = a-X -i if u i (a-X -i ) ≥ pu i (1-X -i ); o Demand x i = 1-X -i otherwise.

Equilibria Three types of equilibria: “Cautious” equilibria: X* = a, certain outcome. “Dangerous” equilibria: X*=1, agents ignore the possibility of a shortage. “Crazy” equilibria: X* > 1, coordination problem.

Cut-offs Proposition 1: Each risk-averse and risk-neutral agent i has a unique cut- off, X i, such that: u i (a-X -i )> p*u i (1-X -i ) if X -i < X i u i (a-X -i ) X i

x2x2 x1x1 X1X1 Best response for 1: a-x 2 Best response for 1: 1-x 2 Best responses for agent 1 1 a 1 a

x2x2 x1x1 Best responses for agent 2 1 a 1 a X2X2 Best response for 2: 1-x 1 Best response for 2: a-x 1

x2x2 x1x1 X2X2 X1X1 Best response for 2: 1-x 1 Best response for 2: a-x 1 Best response for 1: a-x 2 Best response for 1: 1-x 2 Dangerous equilibria Cautious equilibria 1 a 1 a Crazy equilibria 45˚

Equilibria Proposition 2: The game admits at least one non-crazy equilibrium: If ∑ X i < (n-1)a, no cautious equilibrium exists; If ∑ X i > n-1, no dangerous equilibrium exists; If (n-1) a < ∑ X i < n-1, both types of eqs coexist.

Comparative statics As p increases, X i decreases: o the set of dangerous eqs expands o while the set of cautious eqs shrinks. As a increases, X i increases: o the set of dangerous eqs shrinks. o the effect on the set of cautious eqs is ambiguous. If agents become more risk averse: o the set of cautious eqs expands o the set of dangerous eqs shrinks.

Coordinated action Strong Nash equilibria: An equilibrium x is strong if, for any coalition T, and any x’ such that x’ N\T = x N\T : v i (x’) > v i (x) for any i in T v j (x’) < v j (x) for some j in T Coalition-proof equilibria: only self-enforcing deviations.

Coordinated action Theorem 1: All cautious Nash equilibria are strong.

Coordinated action Theorem 2: A dangerous equilibrium, x, is coalition-proof if there does not exist a cautious eq, x’, such that: with δ i ≥0 for all i in T. x’ i = x i – δ i for all i in T x’ i = x i for all i not in T

Coordinated action Corollaries: All cautious equilibria are Pareto efficient and Pareto- dominate many oblivious equilibria. Oblivious eqs are only vulnerable to deviations in which all coalition members reduce their demands The coordination problem can only be solved to the extent that all agents make a simultaneous effort. Remark: No crazy equilibrium is coalition-proof.

Extensions

Extension: multiple thresholds Set of thresholds: r = All cautious eqs are strong. A move from any risky eq. to a « more cautious » eq. is a Pareto improvement. a with prob. (p a ) b with prob. (p b ) … 1 with prob. (1-p a -p b …)

Extension: continuous distributions Uncertainty about threshold: F(r). F(r) is multimodal: experts disagree. v i (x i, X -i )= u i (x i )F(x i +X -i ≤r) If experts disagree sufficiently (« multimodal enough »): multiple equilibria, more and more risky. Any cautious eq. is strong and Pareto efficient. A cautious eq. Pareto dominates any eq. in which no agent consumes less.

Conclusion Simple demand game, introduces threshold effects with uncertainty on the size of the threshold. Cautious and dangerous eqs can coexist even if all are risk averse. Cautious eq. are Pareto efficient and dominates « most » dangerous eqs. Gains from coordinated action can be substantial.

Coexistence of equilibria Even if all agents are risk-averse, oblivious equilibria may exist: Ex: p=0.8, a=0.8, u 1 =u 2 = x ½ (0.4, 0.4) is a cautious eq.: o v i (0.4,0.4)= 0.63 > 0.62 = 0.8*0.6 ½ = v i (0.6,0.4) (0.5, 0.5) is an dangerous eq.: o v i (0.5,0.5)= 0.8*0.5 ½ = 0.56 > 0.54 = v i (0.3,0.5) (1.5, 1.5) is a crazy eq. (and many others) o v i = 0

Proof- Proposition 1 Proof: Define f i (X -i )= u i (a-X -i )- p*u i (1-X -i ). Clearly, f i (a)<0, and f’ i (X -i )=-u’ i (a-X -i ) + p*u’ i (1-X -i ) < 0 by concavity of u -i If f i (0)<0, X i = 0; If f i (0)>0, X i > 0.

Proof- Theorem 1 Sketch of proof by contradiction: For members of coalition T to be better off requires increased demands  outcome no longer certain. Consider an agent j Є T s.t. X’ -j >X -j. She can do no better than to demand 1-X’ -j. However: v j (1-X’ -j,X’ -j ) < v j (1-X -j,X -j ) ≤ v j (a-X -j,X -j ) because x is an equilibrium

Proof- Theorem 2 Sketch of proof: x’ exists  x not coalition-proof. For all i in T, u i (x’ i )≥ p u i (x’ i +1-a) = p u i (x i + ∑δ j ) ≥ p u i (x i ) for all i in T. Thus x’ constitutes a self-enforcing deviation from x.