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Splitting an uncertain (natural) capital. J. Laurent-Lucchetti (U. Bern) J. Leroux and B. Sinclair-Desgagné (HEC Montréal). Introduction. Topic: Strategic behavior under the threat of dramatic events when experts disagree. Issues of climate change: shutdown of thermohaline circulation, - PowerPoint PPT Presentation
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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.