With the conservative approach, we choose the decision which maximises minimum payoff. I was wondering which decision is chosen if 2 decisions have equal minimum payoff (which is the maximum)?
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Problematic is already the use of the words "conservative" and "minimum". There is no absolute minimum, worst case (in continuous models, i.e. reality), as things can always get worse than they have ever been before and than anybody anticipated. Depending on where one defines the minimum, there can always be an even more conservative investor in the market.
Given your tags, let me say that in quantitative areas like portfolio management, if we compare two strategies, e.g. portfolios, for which one risk measure gives equal minimum payoff, then we may apply another risk measure and choose the portfolio which scores better in this risk measure. For example, if a conservative investor considers the robust portfolios with maximized 10%-quantile of weekly returns, and two portfolios are equally good in this measure, he may turn to the one that offers better median returns.