Game Theory, Uncertainty and Asymmetric Information

Simultaneous-move game Nash equilibrium

A Nash equilibrium (NE) is a strategy profile in which no player can raise her own payoff by changing her strategy while the other players keep theirs fixed. In a simultaneous-move game each player chooses without observing the rival's choice, so equilibria are found cell by cell in the payoff matrix. Circle each player's best response to the rival's action; a cell in which both payoffs are circled is a pure-strategy Nash equilibrium (PSNE). A cell in which either player would gain by switching is not an equilibrium.

Key terms

• Best response — the action that gives the highest payoff against a fixed action of the rival. The row player's check compares entries within one column, and the column player's check compares entries within one row.
• Pure and mixed strategy — a pure strategy picks one action with certainty, while a mixed strategy assigns probabilities to actions. A finite game may have no PSNE, but by Nash's theorem it always has an NE in mixed strategies.
• Indifference condition — in a mixed NE each player must be indifferent between the actions she randomises over. The probabilities are set by the rival's payoffs, not by the player's own payoffs.
• Repeated games and trigger threats — when a stage game is repeated, cooperation can be sustained if the discount factor \delta \in (0,1) is high enough. With cooperative payoff \pi_c, one-shot deviation payoff \pi_d, and punishment payoff \pi_p, cooperating forever is better than deviating if \frac{\pi_c}{1-\delta} \ge \pi_d + \frac{\delta\pi_p}{1-\delta}, which is equivalent to \delta \ge \frac{\pi_d-\pi_c}{\pi_d-\pi_p}. The punishment must itself be a stage-game NE so that the threat is credible, which makes the profile a subgame perfect NE (SPNE).
• Several players choosing among tables — when one player chooses which payoff table applies, treat that choice as a strategy. A profile is a PSNE only if no player, including the one choosing the table, gains from a unilateral switch. Count all profiles that pass this test.

Example: Let Row's payoffs be (2,2) and (0,3) in the first row, and (3,0) and (1,1) in the second, with the columns L and R. At (Down, Right) Row gets 1 instead of 0 from Up, and Column gets 1 instead of 0 from Left, so (Down, Right) is the unique PSNE.

Expected utility and risk attitudes

An expected-utility maximiser evaluates a lottery by EU=\sum_i p_i\,u(x_i), where p_i are probabilities and u is the Bernoulli utility function. The expected value of the lottery is E=\sum_i p_i x_i. The certainty equivalent CE is the sure amount with u(CE)=EU, and the risk premium is \pi_R=E-CE. Comparing CE with E classifies the attitude toward risk.

• Risk averse — u is strictly concave, so CE<E and \pi_R>0. By Jensen's inequality E[u(x)]<u(E[x]).
• Risk neutral — u is linear, so CE=E and \pi_R=0. Such a person values a lottery at its expected monetary value, so the maximum price she would pay to play equals E.
• Risk loving — u is strictly convex, so CE>E and \pi_R<0. A person who prefers a sure sum larger than a lottery's expected value is risk loving.
• Hedging — the use of instruments to offset a risk exposure. It describes a behaviour, not a class of preference over lotteries.

Example: A coin toss paying 0 or 100 with equal chance has E=50. With u(w)=\sqrt{w}, EU=0.5(0)+0.5(10)=5, so CE=25 and \pi_R=25>0, which indicates risk aversion.

Arrow–Pratt measures are r_A(w)=-\frac{u''(w)}{u'(w)} and r_R(w)=-\frac{w\,u''(w)}{u'(w)}. For u(w)=\alpha-\beta e^{-rw} with \beta,r>0, the absolute coefficient r_A=r is constant, but r_R=rw increases with wealth. Logarithmic utility \ln w gives r_R=1, which is constant relative risk aversion.

Insurance and asymmetric information

A risk-averse expected-utility maximiser buys insurance to shift wealth from the no-loss state to the loss state. Let the coverage be I, which pays I if the loss occurs, and let the premium be \rho per unit of coverage, so the premium is \rho I. The loss L occurs with probability \pi. Final wealth is W_L=W-L+I-\rho I in the loss state and W_N=W-\rho I otherwise.

The optimal coverage solves the first-order condition obtained by differentiating \pi\,u(W_L)+(1-\pi)\,u(W_N) with respect to I. The condition is \pi(1-\rho)\,u'(W_L)=(1-\pi)\,\rho\,u'(W_N), or equivalently \frac{u'(W_L)}{u'(W_N)}=\frac{\rho(1-\pi)}{\pi(1-\rho)}.

• Actuarially fair premium — when \rho=\pi, the condition gives u'(W_L)=u'(W_N). With a strictly concave u, this implies W_L=W_N, so the agent fully equalises wealth across states.
• Log utility — with u=\ln w, u'(w)=1/w, so the condition becomes \frac{W_N}{W_L}=\frac{\rho(1-\pi)}{\pi(1-\rho)}. Solving this equation for I is the standard method for insurance-coverage problems.
• Asymmetric information — when the insurer cannot observe the buyer's risk type, the premium is set on the pool. High-risk buyers are more likely to purchase, which is adverse selection. Hidden actions that change the risk after coverage is bought are moral hazard. Both break the link between the premium and the buyer's true risk.

Battle of the sexes mixed strategy Nash equilibrium

A Battle of the Sexes game has players who want to coordinate but disagree on which joint action is better. It has two pure-strategy NE, in which both players coordinate on the point preferred by one of them. It also has a mixed NE in which each player randomises so that the rival is indifferent between her two actions.

Let p be the probability that player 1 plays F, and q the probability that player 2 plays F. Player 2's indifference condition uses player 1's payoffs and determines q, while player 1's indifference condition uses player 2's payoffs and determines p. For a 2×2 coordination game with player 1 payoffs x_F, x_C and player 2 payoffs y_F, y_C at the two coordinated points, the mixed NE satisfies $p=\frac{y_C}{y_F+y_C}, \qquad q=\frac{x_C}{x_F+x_C}.$