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Prove that Choice Coherence implies Independence of Irrelevant Alternatives (IIA).

From the definition of choice coherence, we have this: A choice function c satisfies choice coherence if, for every pair x and y from X and A and B from A, if x, y are in A and B, x in c(A), and y not in c(A), then y is not in c(B).

From the definition of IIA, we have this: If A is preferred to B out of the choice set {A,B}, introducing a third option X, expanding the choice set to {A,B,X}, must not make B preferable to A.

The converse is not true (IIA does not imply choice coherence) since we can construct a counterexample, as per here:

C({x,y,z}) = {x}

C({x,y}) = {x}

C({x}) = {x}

C({y}) = {y}

C({x,y}) = {x,y}

Since IIA is the weaker assumption of the two, how can we show that choice coherence implies IIA? Thanks for your help.

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I think your notation makes this more confusing that is necessary since $A$ and $B$ represent sets of alternatives in one setting and alternatives in the other setting.

So write IIA as:

If $x\in C\big(\{x,y\}\big)$ and $y\notin C\big(\{x,y\}\big)$, then $y\notin C\big(x,y,z\big)$.

To prove this from choice coherence, simply choose $A$ and $B$ appropriately.

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