Repeated games add depth to strategic interactions by playing the same game multiple times. This section explores finite and infinite horizon games, highlighting how the number of repetitions affects player strategies and outcomes.
Discount factors, subgame perfect equilibrium, and strategies like grim trigger are key concepts. These tools help analyze how players balance short-term gains against long-term cooperation in repeated interactions.
Finite and Infinite Horizon Games
Repeated Games and Their Horizons
- Repeated games involve playing the same game (stage game) multiple times
- Finite horizon repeated games have a known, fixed number of repetitions
- Players are aware of when the game will end
- Backward induction can be used to solve for the subgame perfect equilibrium
- Infinite horizon repeated games continue indefinitely with no known end point
- Future payoffs are discounted by a discount factor $\delta$ (between 0 and 1)
- Higher discount factors indicate more patient players who value future payoffs more
Discount Factors in Infinite Horizon Games
- The discount factor $\delta$ represents how much players value future payoffs relative to current ones
- Ranges from 0 to 1, with higher values indicating more patient players
- $\delta = 0$: Players only care about immediate payoffs (equivalent to one-shot game)
- $\delta = 1$: Players value future payoffs equally to current ones (perfectly patient)
- Discount factors can vary between players, reflecting different levels of patience
- The discounted sum of future payoffs is calculated as $\sum_{t=0}^{\infty} \delta^t \pi_t$, where $\pi_t$ is the payoff in period $t$
Equilibrium Concepts
Subgame Perfect Equilibrium and Backward Induction
- Subgame perfect equilibrium (SPE) is a refinement of Nash equilibrium for dynamic games
- A strategy profile is an SPE if it represents a Nash equilibrium of every subgame of the original game
- Eliminates non-credible threats and ensures strategies are optimal at every decision point
- Backward induction is a technique used to solve for SPE in finite horizon games
- Start at the final period and determine optimal actions
- Work backwards, period by period, using future optimal actions to determine current optimal actions
- Ensures players' strategies are sequentially rational and credible
Continuation Values in Repeated Games
- The continuation value is the expected discounted sum of future payoffs from a given point onward
- Represents the value of the game from a particular decision node, assuming optimal play
- Used to determine optimal actions at each decision point in repeated games
- Players compare immediate payoffs plus continuation values to determine best responses
- In infinite horizon games, continuation values are used to calculate the discounted sum of future payoffs
- $V_i = (1-\delta) \pi_i + \delta V_i$, where $V_i$ is player $i$'s continuation value and $\pi_i$ is their stage game payoff
Strategies in Repeated Games
Grim Trigger Strategy
- Grim trigger is a strategy in repeated games that incentivizes cooperation through punishment threats
- Players start by cooperating and continue to do so as long as all players have cooperated in the past
- If any player defects, the grim trigger player permanently switches to defection for all future periods
- Acts as a severe punishment mechanism to deter defection and sustain cooperation
- Effectiveness depends on the discount factor and stage game payoffs
- Higher discount factors make the threat of future punishment more severe
- Grim trigger can sustain cooperation in the Prisoner's Dilemma if $\delta \geq \frac{T-R}{T-P}$, where $T$, $R$, and $P$ are the temptation, reward, and punishment payoffs
- Limitations include its harshness (one defection leads to permanent punishment) and lack of forgiveness
- Strategies like tit-for-tat (copying the opponent's previous action) can be more effective in noisy environments