Question

Consider a two-player simultaneous move game with \(S_{1}=S_{2}=[0,1]\). Suppose that, for \(i=1,2\), player \(i\) 's best response can be represented as a function of \(\bar{s}_{j}\), the expected value of \(s_{j}\) according to player \(i\) 's belief (where \(j=-i\) ). Specifically, player 1's best response function is given by \(B R_{1}\left(\bar{s}_{2}\right)=\bar{s}_{2}\), and player 2 's best response function is given by $$ B R_{2}\left(\bar{s}_{1}\right)=\min \left\\{\frac{1}{4}+\bar{s}_{1}, 1\right\\} . $$ To be clear about this functional form, if \(\frac{1}{4}+\bar{s}_{1}<1\), then player 2 's best response is \(\frac{1}{4}+\bar{s}_{1}\). Otherwise, player 2's best response is 1 . (a) Graph the best response functions for each player. (b) Find \(B_{1}\) and \(B_{2}\). Calculate the rationalizable strategy set and explain which strategies are removed in each round of the iterated dominance procedure.

Short answer

The rationalizable strategy set is \((1/3,1/3)\), where both players match each other's strategy of 1/3.

Step-by-step solution

Graph Player 1's Best Response

Player 1's best response function is given by \( BR_1(\bar{s}_2) = \bar{s}_2 \). Since \( S_2 = [0,1] \), this means that Player 1's best response to any strategy of Player 2 is to match it exactly. The graph of this function is a 45-degree line intersecting the origin with a slope of 1. It runs from (0,0) to (1,1) on the square \([0,1] \times [0,1]\).

Graph Player 2's Best Response

Player 2's best response function is given by \( BR_2(\bar{s}_1) = \min \left\{\frac{1}{4} + \bar{s}_1, 1\right\} \). For \( \bar{s}_1 \leq \frac{3}{4} \), Player 2's response is \( \frac{1}{4} + \bar{s}_1 \), forming a line starting at (0,0.25) and increasing with slope 1. Once \( \bar{s}_1 \) exceeds \( \frac{3}{4} \), the response caps at 1, thus forming a horizontal line from (0.75,1) to (1,1).

Identify Intersection of Best Responses

To find the rationalizable strategy set, we find where \( BR_1 \) and \( BR_2 \) intersect: set \( BR_1(\bar{s}_2) = BR_2(\bar{s}_1) \). Solve \( \bar{s}_2 = \frac{1}{4} + \bar{s}_1 \) where \( \bar{s}_1 \leq \frac{3}{4} \). Since \( \bar{s}_1 = \bar{s}_2 \), the solution is \( \bar{s}_1 = \bar{s}_2 = 1/3 \). Beyond \( \bar{s}_1 = 3/4 \), intersections cannot be improvements for Player 2.

Determine Rationalizable Strategy Sets

The intersection at \((1/3,1/3)\) indicates a mutual best response in \([0,3/4]\). Initially, rationalizable strategies for both players are in [0,1]. By iterated dominance, the strategy set progressively shrinks as non-best responses are eliminated from consideration. Strategies in intervals \([0,1/3)\) and \((1/3,1]\) become strictly dominated by \(1/3\) for both players.

Key concepts

Best Response Function

The best response function is essential in game theory as it explains how a player chooses their optimal strategy based on expectations of the other player's strategy. In this game, both players have specific best response functions that depend on their beliefs about the opponent's strategy. For Player 1, the best response function is simple: whatever they believe Player 2 is doing, they will match it exactly, illustrated by the function \( BR_1(\bar{s}_2) = \bar{s}_2 \). Player 1 will try to mirror Player 2's expected choice.
On the other hand, Player 2's best response function, \( BR_2(\bar{s}_1) = \min \left\{\frac{1}{4} + \bar{s}_1, 1\right\} \), reflects a more cautious approach. It means Player 2 will add a fixed value (0.25) to their expectation of Player 1's choice, but they will cap their response at 1. This approach limits extreme responses and stabilizes their action regardless of any substantial increase in Player 1's strategies.

Simultaneous Move Game

In a simultaneous move game, both players choose their actions without knowing the other's choice. This type of game captures a sense of suspense and strategic uncertainty since neither player sees the other's move. Each player's decision hinges on their belief about the other's strategy.
In the exercise scenario, both players have a set of strategies represented by the interval \([0, 1]\). They decide simultaneously based on their best response functions. The simultaneous move game demands each player rationalize over this uncertainty, making predictions and reacting according to their best responses.

• This approach closely mimics real-life situations where decisions are made concurrently and independently.
• Understanding simultaneous move games can help when analyzing scenarios like bidding in auctions or setting prices in competitive markets.

Rationalizable Strategy Set

The rationalizable strategy set in game theory involves all strategies that can be derived from rational thought processes about others' strategies. In essence, it's a range of strategies that make logical sense when players consider what the other might do.
In this game, we found the strategies that intersect as both players try to best respond to each other's beliefs. By analyzing both players' best responses, we determine the intersection is at \((1/3, 1/3)\). This point signifies mutual optimal strategies within the range \([0, 3/4]\), which means both players act rationally based on available information.
The consequence of finding the rationalizable strategy set is that it helps players avoid improbable or non-optimal strategies and focus only on those strategies that can potentially lead to a beneficial outcome for all involved.

Iterated Dominance Procedure

The iterated dominance procedure aims to narrow down strategy options by successively eliminating dominated strategies—those which are not optimal responses. It simplifies decision-making by focusing only on the best strategies.
In our exercise, starting with the full strategy space \([0, 1]\), strategies not aligning with the best response functions get discarded. Using iterated dominance, strategies less than \(1/3\) and more than \(1/3\) but not yielding a better result than \(1/3\) are eliminated.
This step-by-step culling results in the focal point strategy \(1/3\), which both players are unlikely to deviate from due to its rational advantage. This procedure is practical as it primes players to focus on realistic strategies, eliminating unnecessary complexity and enhancing predictive understanding of others' likely choices.

• This method guarantees the remaining strategies are those that withstand rational examination rounds.
• It systematically breaks down broader strategy choices into manageable and more predictable sets for decision makers.