Risk-Aversion under External Uncertainty

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Risk-Aversion under External Uncertainty

• Players: One decision maker
• Strategies: Choose the payoff-dominant risky option; Choose the risk-dominant agreement.

Rules

• Start with a risky payoff-dominant option and a certain risk-dominant agreement; The decision maker chooses between the risky option and the certain agreement.
• The player who maximises expected utility chooses the preferred option.
• The decision maker values money using the utility function $u(x)=\sqrt{x}$; The payoff-dominant option gives $160000$ in the good state and $40000$ in the bad state.
• The risk-dominant agreement gives $100000$ in both states.
• Each state occurs with probability $\frac{1}{2}$.

Payoff Details

& Good & Bad \ Payoff-dominant option & $160000$ & $40000$ \ Risk-dominant agreement & $100000$ & $100000$

Derivation (Best Response Analysis)

• Expected utility from the payoff-dominant risky option is:

• Expected utility from the risk-dominant agreement is:

• The risk-dominant agreement gives higher expected utility.

Bernoulli Utility Function

Definition:

A Bernoulli utility function maps monetary payoff $x$ into utility $u(x)$.

Risk aversion is represented by concavity:

For a risky payoff $X$,

• Here,

• The risky option has

• Utility of expected wealth:

• Certainty equivalent:

• Risk premium:

Diagram (Bernoulli Utility)

Derivation (Nash Equilibrium)

• With concave utility, the decision maker dislikes payoff dispersion.
• The risky option and the agreement have the same expected monetary payoff.
• The agreement is preferred because it gives the same payoff in both states.

Nash Equilibrium

Result:

The risk-averse decision maker chooses the risk-dominant agreement because $100\sqrt{10}>300$.

Social Optimum

• Expected monetary payoff is unchanged.
• Expected utility is higher under the risk-dominant agreement.
• The optimal choice for a risk-averse decision maker is the certain payoff.

Insights

Insight:

• Risk aversion is about utility, not only money.
• Concave utility makes equalised payoffs more attractive.
• A certain payoff can be better than a risky payoff with the same average value.