Investment Game with Moral Hazard

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Investment Game with Moral Hazard

• Players: Player 1 is the Investor; Player 2 is the Friend.
• Strategies: Each period, the Investor chooses Invest or Safe; if investment occurs, the Friend chooses Repay or Abscond.
• Rules: The game repeats infinitely. Future payoffs are discounted by $\delta=\frac{1}{1+r}$.

Payoff Details

• If the Investor invests $100$, total output is $130$.
• If the Friend repays $120$, payoffs are $(120,10)$.
• If the Friend absconds with $130$, payoffs are $(0,130)$.
• If the Investor chooses the safe outside option, payoffs are $(100(1+r),0)$.

Game Tree

Derivation (Grim Trigger and One-Shot Deviations)

Grim trigger strategies

• Investor: invest in period $0$ and continue investing after histories with only Invest, Repay; after any deviation, choose Safe forever.
• Friend: repay after every investment as long as all previous investments were repaid; after any deviation, abscond if investment is ever offered.
• On the equilibrium path, play is Invest, Repay forever.
• After a deviation, the punishment path is Safe forever, giving payoffs $(100(1+r),0)$ each period.

Investor no-deviation condition

• If the Investor follows grim trigger, the continuation payoff is

• If the Investor deviates once to the safe outside option, grim trigger gives the safe payoff forever:

• No one-shot deviation requires

• Simplify the geometric sums:

• Since $1-\delta>0$,

• Divide by $100$:

• Therefore,

Insight:

The Investor invests only if repayment beats the safe outside return. This constraint compares two permanent payoff streams, so patience cancels out.

Friend no-deviation condition

• If the Friend follows grim trigger, the continuation payoff is

• If the Friend deviates once by absconding, the Friend gets $130$ immediately and $0$ forever after:

• No one-shot deviation requires

• Simplify the geometric sums:

• Multiply by $1-\delta>0$:

• Expand:

• Rearrange:

• Divide by $130$:

• Substitute $\delta=\frac{1}{1+r}$:

• Since $1+r>0$, multiply both sides by $13(1+r)$:

• Expand:

• Rearrange:

• Therefore,

Insight:

The Friend repays only if the future stream of $10$ repayments is valuable enough to outweigh the immediate temptation to steal $130$.

Punishment-state deviations

• In the punishment state, the Investor receives $100(1+r)$ forever by choosing Safe.
• A one-shot investment during punishment gives at most $0$ immediately if the Friend absconds, so it is not profitable.
• The Friend receives $0$ when no investment occurs and has no profitable one-shot deviation on the punishment path.

Nash Equilibrium

Result:

By the one-shot deviation principle, grim trigger is a subgame perfect Nash equilibrium if

and

The binding constraint is the Friend’s no-deviation condition, since

Therefore the final equilibrium condition is

Social Optimum

• Perpetual investment and repayment generates total surplus $130$ each period.
• The safe outside option generates total surplus $100(1+r)$ each period.
• Under the equilibrium condition $r\leq \frac{1}{12}$, investment is socially efficient because $130>100(1+r)$.

Insights

Insight:

The investment relationship fails first because of the Friend’s moral hazard, not because of the Investor’s outside option. A higher interest rate lowers patience and makes the future relationship less able to discipline absconding.