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if both situated at 1/2 in [0,1] and suppose A moves right to 3/4 B is still at half..draw the situation. B now sells to 50% buyers + half of those situated between 1/2 and 3/4 ie 12.5% = 62.5% of buyers but A sells to only 37.5%...this holds for movement anywhere left or right from 1/2 point ...so basically neither of the two hav incentive to deviate from 1/2 cause the one who does his market share falls..
if A at 1/4 B at 3/4 then A has incentive to move right as his mkt sh rises as he does so but only upto 1/2 , same for B..same logic for 0 and 1 and 1/3 and 2/3
so ultimately both end up at 1/2
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