Game Theory - Rationalizability
Introduction to Rationalizability in Game Theory
Overview of Rationalizability
- The concept of rationalizability assumes three key principles: players form beliefs, they best respond to these beliefs, and both aspects are common knowledge among players.
Example: Ice Cream Vendors Location Game
- The example involves two ice cream vendors who must independently choose one of nine beach regions for their location while applying for permits.
- Each region has 50 consumers willing to buy ice cream, with vendors earning a payoff of one per consumer. Consumers will walk to the closest vendor.
Payoff Structure and Strategy Space
- The total potential payoff across all regions is 450 (9 regions x 50 consumers). If both vendors are equidistant from a region, consumers split equally between them.
- For instance, if Vendor One is in region three and Vendor Two in region eight, Vendor One earns a total of 250 by attracting consumers from five regions.
Dominated Strategies
- Certain strategies can be deemed dominated; specifically, locating in regions one or nine is less advantageous than choosing regions two or eight.
- Analyzing payoffs shows that Player One always benefits more by choosing region two over region one against any strategy Player Two might employ.
Iterative Elimination of Dominated Strategies
- Through iterative reasoning, the set of viable strategies narrows down from 1,...,9 to 2,...,8, then further reduces through repeated rounds until only region five remains as the unique prediction.
Understanding the 2/3 Game and Rationalizability
The Concept of the 2/3 Game
- The 2/3 game involves players choosing integers between 1 and 100, aiming to be closest to two-thirds of the average choice. This concept diverges from traditional voting models where players tend to converge towards a median position.
- Players' payoffs are structured such that they receive higher rewards for being closer to two-thirds of the average, which is calculated based on all players' choices.
Payoff Structure and Dominance
- Player i's payoff depends on their choice relative to others, specifically how close they are to 2/3 times textaverage . If player i is closest, they earn 100/M , where M is the number of players also closest.
- It’s challenging to prove that choosing 100 is dominated because it may not always yield a lower payoff than choosing 99 depending on other players’ choices.
Best Response Dynamics
- Choosing 99 becomes a better response if everyone else chooses numbers like 100 since it positions player i closer to two-thirds of the average.
- Through iterative reasoning, as rationality rounds progress, players will eventually gravitate towards lower numbers until potentially reaching one. However, in practice, results often cluster around values in the twenties due to strategic uncertainty among players.
Common Knowledge and Rationalizability
- The complexity arises from common knowledge; each player's belief about others' rationality affects their own strategy. This leads to deviations from theoretical predictions in actual gameplay scenarios.
- Despite its complexities, rationalizability remains a robust solution concept but requires careful consideration regarding its applicability in different contexts.
Matrix Example: Dominated Strategies
- A matrix example illustrates dominance concepts where player two's strategies can be evaluated against player one's actions.
- By analyzing payoffs under various conditions (e.g., when Z or Y is played), we can determine which strategies are dominated and thus eliminated from consideration.
Iterated Deletion of Dominated Strategies
- Player one recognizes that certain strategies (like Z and A) are dominated by others based on probability weights assigned to outcomes.
- The process continues until no further dominated strategies exist; this method highlights how iterated deletion can simplify decision-making in finite games without yielding unique predictions.