Question:
Why is there need for transitivity assumption to develop models? Can't we develop models without the transitivity assumption?Answer:
Without the transitivity assumption, an economic agent can exhibit preference cycles, leading to potential risk-free arbitrage opportunities.
For instance, consider Theo, who prefers chicken to steak and steak to pizza. Transitivity would imply that Theo also prefers chicken to pizza. However, if transitivity is violated, Theo might prefer pizza to chicken, creating a strict preference cycle: chicken is preferred to steak, steak to pizza, and pizza to chicken.
This cycle might become a problem because it can generate risk-free arbitrage. Let’s introduce Sophie, who identifies this cycle. Suppose Theo currently has chicken for dinner. Sophie offers to trade Theo’s chicken for pizza, and since Theo strictly prefers pizza, he agrees and might even pay Sophie a small fee (e.g., one cent) for the trade. Now Theo has pizza. Sophie can then offer to trade the pizza for steak, which Theo also prefers, and he agrees again. Finally, Sophie offers to trade the steak for chicken, which Theo prefers, completing the cycle.
After these trades, Theo is back to having chicken but has paid Sophie one cent. If Sophie repeats this cycle a million times, she can extract $10,000 from Theo. Hopefully, Theo would recognize the flaw in his non-transitive preferences and correct it before getting too poor.
