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

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: xi ≥ 0.

• If ∑xi ≤ r, each agent receives xi. 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: ui(xi).

• ui’s are concave (risk aversion and risk neutrality) non-decreasing and ui(0)=0.

• Expected payoff of an agent i:

vi (xi, X-i)= ui(xi) Ι(X≤a)+p ui(xi) Ι(a<X≤1)

where X = ∑xi = xi+X-i

Best responsesFor each agent i:

• If X-i > a:

o If X-i < 1: demand xi = 1-X-i

o If X-i ≥ 1 : demand xi = 0 or anything higher.

• If X-i ≤ a:

o Demand xi = a-X-i if ui(a-X-i ) ≥ pui(1-X-i );

o Demand xi = 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, Xi, such that:

ui (a-X-i)> p*ui(1-X-i) if X-i < Xi

ui (a-X-i)< p*ui (1-X-i) if X-i > Xi

x2

x1

X1

Best response for 1: a-x2

Best response for 1: 1-x2

Best responses for agent 1

1a

1

a

x2

x1

Best responses for agent 2

1a

1

a

X2

Best response for 2: 1-x1Best response for 2: a-x1

x2

x1X2

X1

Best response for 2: 1-x1Best response for 2: a-x1

Best response for 1: a-x2

Best response for 1: 1-x2

Dangerous equilibria

Cautious equilibria

1a

1

a

Crazy equilibria

45˚

EquilibriaProposition 2:The game admits at least one non-crazy equilibrium:

• If ∑ Xi < (n-1)a, no cautious equilibrium exists;

• If ∑ Xi > n-1, no dangerous equilibrium exists;

• If (n-1) a < ∑ Xi < n-1, both types of eqs coexist.

Comparative statics• As p increases, Xi decreases:

o the set of dangerous eqs expands o while the set of cautious eqs shrinks.

• As a increases, Xi 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 expandso 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 = xN\T:

vi(x’) > vi(x) for any i in T vj(x’) < vj(x) for some j in T

• Coalition-proof equilibria: only self-enforcing deviations.

Coordinated action

Theorem 1:All cautious Nash equilibria are strong.

Coordinated actionTheorem 2:  A dangerous equilibrium, x, is coalition-proof if there does not exist a cautious eq, x’, such that:

with δi≥0 for all i in T.

x’i = xi – δi for all i in T

x’i = xi for all i not in T

Coordinated actionCorollaries:

• 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. (pa)

b with prob. (pb)…1 with prob. (1-pa-pb…)

Extension: continuous distributions• Uncertainty about threshold: F(r).

• F(r) is multimodal: experts disagree.

vi(xi, X-i)= ui(xi)F(xi+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 equilibriaEven if all agents are risk-averse, oblivious equilibria may exist:• Ex: p=0.8, a=0.8, u1=u2= x½

• (0.4, 0.4) is a cautious eq.: o vi(0.4,0.4)= 0.63 > 0.62 = 0.8*0.6 ½ = vi(0.6,0.4)

• (0.5, 0.5) is an dangerous eq.: o vi(0.5,0.5)= 0.8*0.5 ½ = 0.56 > 0.54 = vi(0.3,0.5)

• (1.5, 1.5) is a crazy eq. (and many others) o vi = 0

Proof- Proposition 1

Proof:

Define fi(X-i)= ui (a-X-i)- p*ui(1-X-i).

Clearly, fi(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 fi(0)<0, Xi = 0;• If fi(0)>0, Xi > 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:

vj(1-X’-j,X’-j) < vj(1-X-j,X-j) ≤ vj(a-X-j,X-j)

because x is an equilibrium

Proof- Theorem 2Sketch of proof: x’ exists x not coalition-proof.

For all i in T,

ui(x’i)≥ p ui(x’i+1-a)

= p ui(xi + ∑δj )

≥ p ui(xi) for all i in T.

Thus x’ constitutes a self-enforcing deviation from x.