Question

Consider a repeated game between a supplier (player 1) and a buyer (player 2). These two parties interact over an infinite number of periods. In each period, player 1 chooses a quality level \(q \in[0,5]\) at cost \(q\). Simultaneously, player 2 decides whether to purchase the good at a fixed price of 6 . If player 2 purchases, then the stage-game payoffs are \(6-q\) for player 1 and \(2 q-6\) for player \(2 .\) Here, player 2 is getting a benefit of \(2 q\). If player 2 does not purchase, then the stage-game payoffs are \(-q\) for player 1 and 0 for player 2 . Suppose that both players have discount factor \(\delta\). (a) Calculate the efficient quality level under the assumption that transfers are possible (so you should look at the sum of payoffs). (b) For sufficiently large \(\delta\), does this game have a subgame perfect Nash equilibrium that yields the efficient outcome in each period? If so, describe the equilibrium strategies and determine how large \(\delta\) must be for this equilibrium to exist.

Short answer

(a) Efficient quality level is 5. (b) Yes, if \(\delta \geq 0.2\).

Step-by-step solution

Determine Payoffs Sum

To calculate the efficient quality level, we need to maximize the sum of payoffs when Player 2 purchases: \((6-q) + (2q-6) = q\). Thus, we focus on maximizing \(q\) given the constraints of quality level \(q \in [0,5]\). Hence, the efficient quality level is \(q = 5\).

Check If Efficient Outcome Can Be Sustained

To sustain the efficient outcome \(q=5\) in each period, we need a strategy that discourages deviation by either player. Since player 1 would rather maximize just by choosing \(q=0\) to save on cost unless disciplined by the threat of future non-purchases, we consider possible equilibria.

Analyze Deviation for Player 1

If player 1 deviates from \(q=5\) to \(q=0\), they receive payoff \(6\) once, with future payoffs being just \(-q\) if player 2 refuses to buy. For \(q=5\), the payoff each period is \(6-5=1\). Hence, player 1's discounted sum of future payoffs if adhering to \(q=5\) is \(\frac{1}{1-\delta}\).

Analyze Deviation for Player 2

For player 2, the payoff is \(4\) for purchasing \(q=5\), with zero payoff if deviating (not buying). In order for not deviating, the comparison is \(\frac{4}{1-\delta}\) versus temporary deviation followed by payoff \(0\).

Sufficient Delta Calculation

Both players must find the future payoffs so valuable that no one deviates. For Player 1, if adhering gives \(\frac{1}{1-\delta}\) is greater than \(6+0\), Player 2 will similarly comply. Inequality \(\delta \geq 0.2\) ensures both adhere.

Key concepts

Game Theory

Game theory is a mathematical framework used to analyze strategic interactions between rational decision-makers, often called players. Here, the supplier (player 1) and the buyer (player 2) engage in a repeated interaction, where each player selects strategies to maximize their payoff over time. Game theory examines these scenarios by considering how players anticipate responses and optimize their actions accordingly.

Key elements of these analyses include:

• Strategies: Potential choices the players can make in each stage.
• Payoffs: The outcomes received from a combination of chosen strategies, which depend not only on one player's action but also on the other's decision.
• Equilibrium: A solution concept where players have no incentive to unilaterally deviate from their chosen strategies.

In this repeated game, player 1's strategy involves choosing a quality level, while player 2 decides on purchasing the product. The game's dynamics evolve over countless periods, allowing for strategy adjustments to exploit observed behaviors.

Subgame Perfect Nash Equilibrium

The Subgame Perfect Nash Equilibrium (SPNE) is a refinement of Nash Equilibrium suited for dynamic games, particularly those with sequential or repeated plays. It requires that players maintain optimal strategies at every point within the game, unlike a general Nash equilibrium that only needs to satisfy the condition at the onset.

In this exercise, the SPNE tackles how both players maintain their strategies across numerous periods. An equilibrium is considered subgame perfect if it constitutes a Nash Equilibrium in each subgame of the original game. This means:

• Player 1 should consistently choose the quality level that maximizes their future discounted payoffs.
• Player 2 should carry out purchasing decisions that ensure long-term optimal benefit, discouraging player 1 from selecting inferior strategies.

Reaching SPNE ensures that neither player benefits from changing their strategy at any stage of the infinite game, leading to long-term sustainability of the desired outcomes—like always choosing the efficient quality level.

Discount Factor

The discount factor, often represented as \( \delta \), captures how players value future payoffs versus immediate ones. A higher \( \delta \) indicates that a player values future rewards almost as much as current rewards, while a lower \( \delta \) shows a preference for short-term gains over long-term benefits.

In the repeated game of supplier and buyer, this concept plays a pivotal role:

• For both players to adhere to a strategy ensuring efficient outcomes, \( \delta \) needs to be sufficiently large, meaning future considerations outweigh immediate deviations.
• If \( \delta \) is too low, a player might break from optimal strategies in the short-term, seeking immediate, higher payoffs instead of waiting for collective future benefits.

To maintain the efficient outcome \( q=5 \), the exercise calculates a necessary \( \delta \) value that makes future cooperation more appealing than deviating. This reinforces adherence to strategies leading to the Subgame Perfect Nash Equilibrium.

Efficient Outcome

An efficient outcome in game theory is a strategy profile that maximizes total payoffs across all players involved. It considers the sum of all individual payoffs, directing players toward cooperative strategies that benefit all parties rather than just focusing on individual gain.

In the supplier-buyer game, the efficient quality level is determined by maximizing the total benefit when purchasing occurs: \( 6-q + 2q-6 = q \). Thus, choosing \( q = 5 \) ensures both players achieve maximum collective payoffs.

• The supplier (player 1) delivers high quality, aligning with player 2's interest, thus benefiting both.
• Although this might not be individually optimal for the supplier considering cost alone, the sum of payoffs dictates this choice.

Achieving such outcomes is vital as these cooperative moves help sustain beneficial interactions and ensure the overall success of strategic engagements in repeated games.