Discounting

Game Theory · Axiom Academy

Understanding how players value future payoffs in repeated games The discount factor δ (delta) represents how much a player values future payoffs relative to present payoffs. It ranges from 0 to 1: Definition: If a player receives payoff π in the next period, its present value is δπ . Future payoffs are nearly as valuable as current payoffs. The player cares deeply about long-term consequences. Future payoffs are heavily discounted. The player strongly prefers immediate rewards. Watch the animation to see how different discount factors affect the present value of a future payoff: 2. Present Value of Payoff Streams In repeated games, players receive a sequence of payoffs over time: π 0 , π 1 , π 2 , ... The present value of this entire stream is: Each period's payoff is discounted by δ raised to the power of how many periods in the future it occurs: Today (t=0): Payoff π 0 has present value π 0 Next period (t=1): Payoff π 1 has present value δπ 1 Two periods ahead (t=2): Payoff π 2 has present value δ²π 2 The animation shows how payoffs in different periods contribute to total present value: 3. Constant Payoffs and Geometric Series A common scenario in repeated games: a player receives the same payoff π in every period forever. What is the present value of this infinite stream? This is a geometric series with first term π and ratio δ. Using the formula for an infinite geometric series (when |δ| < 1): Key Formula: The present value of receiving payoff π forever is π/(1-δ)

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