Indifferent choice assumption in economics Take, for instance, that you have a game similar to the one proposed by Levitt and Duggan (2000) in sumo wrestling (link: http://www.nber.org/papers/w7798.pdf, section II (pages 9-10) in which the paper describes the model). 

Summarizing, a fighter maximizes a expected return in a non-rigged match. For a given fighter consider taking the bribe, the other minimizes a side payment (bribe) to the point of indifference between taking and not taking the bribe -- so, why would the fighter setting the side payment would not set the bribe so it would be marginally better to accept the bribe than not taking it? Why do the authors include the point of indifference among the possible solutions?
Because otherwise you would not have subgame perfect equilibria. Suppose I have an item you need and the item is worthless to me. I would be happy to give you the item for any positive amount of money. If you give me no money for it, I am indifferent, both giving you the item and not giving you the item is an optimal choice for me. Just to be clear: we are not bargaining. You set I price and I will take it or leave it. 

If you offer price 0 and I give you the item that is an equilibrium. No one could have made a better choice given the other person's choice.
 If you offer price 0 and I do not give you the item that is not an equilibrium, because I would give you the item for any small positive payment and you would be willing to pay that, so you could have made a better move.
 If you offer price $2a&gt;0$ I will give you the item. But this is not an equilibrium. You could also have offered me $a$, which is also positive, and I would also have given you the item then. So you could have saved some money, your original move was not optimal.

Because there is no smallest positive number, the only equilibrium in such cases is when one side is indifferent.