Risk-Sharing under External Uncertainty

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Risk-Sharing under External Uncertainty

• Players: Two players, Agent 1 and Agent 2.
• Strategies: No agreement: each agent keeps their realised payoff; Risk-sharing agreement: the lucky agent transfers $60000$ to the unlucky agent; Insurance: the agent pays a premium to remove the risky payoff.

Rules

• Start with two agents facing negatively correlated risky outcomes; Players choose no agreement, risk sharing, or insurance arrangements.
• The player who smooths consumption when utility is concave improves expected utility.
• The agents are engaged in similar activities in different geographical areas; Their outcomes are negatively correlated: if things go well for one agent, they go badly for the other.
• Each agent faces a good outcome and a bad outcome with equal probability.
• Without agreement, the good state pays $160000$ and the bad state pays $40000$; With risk sharing, the lucky agent pays $60000$ to the unlucky agent.

Payoff Details

& Good & Bad \ No agreement & $160000$ & $40000$ \ Probability & $0.5$ & $0.5$ \ Risk-sharing agreement & $100000$ & $100000$ \ Probability & $0.5$ & $0.5$

Derivation (Best Response Analysis)

• Without agreement, expected monetary payoff is:

• With risk sharing, expected monetary payoff is:

• Expected value is unchanged.
• Risk is eliminated because each agent receives $100000$ for sure.
• A risk-neutral agent is indifferent between the two arrangements.
• A risk-averse agent strictly prefers the risk-sharing agreement.

Utility and Risk Aversion

• Suppose utility is:

• Expected utility without risk sharing is:

• Expected utility with risk sharing is:

• The same expected monetary payoff gives higher utility when risk is removed.

Insurance

• Insurance companies take on risk in exchange for payment.
• The certainty equivalent is the guaranteed payoff giving the same utility as the risky lottery.
• Since $u(90000)=300$, receiving $90000$ for sure gives the same utility as the risky payoff.
• Maximum willingness to pay for full insurance is:

• The agent is willing to pay up to $10000$ to eliminate the risk.

Graph Interpretation

• The blue curve is utility, $u(x)=\sqrt{x}$.
• The yellow chord is the linear combination of utilities across risky states.
• The chord connects $(40000,u(40000))$ and $(160000,u(160000))$.
• Its equation is:

• If the good state occurs with probability $p$, expected wealth is:

• Substituting this into the chord gives:

• At $x=100000$, the chord gives expected utility $E[u(x)]=300$.
• The curve gives utility of expected wealth, $u(E[x])=u(100000)=100\sqrt{10}$.
• Since utility is concave:

Derivation (Nash Equilibrium)

• If both agents are risk averse, both prefer risk sharing to no agreement.
• The agreement is mutually beneficial because each agent gets the expected payoff for sure.
• The agreement works because their risks move in opposite directions.

Nash Equilibrium

Result:

For risk-averse agents, the risk-sharing agreement is the stable outcome: the lucky agent pays $60000$ to the unlucky agent, and both receive $100000$ for sure.

Social Optimum

• Expected total payoff is unchanged by the transfer.
• Welfare rises for risk-averse agents because uncertainty is removed.
• The social optimum is full risk sharing: both agents receive the certain payoff $100000$.

Insights

Insight:

• Risk sharing eliminates uncertainty without changing expected value.
• Risk-averse individuals prefer certainty.
• Insurance pricing reflects the gap between expected value and certainty equivalent.
• Expected utility lies on the chord, not on the utility curve.