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4 votes
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Mixed strategy in extensive form games with complete and perfect information

I assume that implicit in the lemma is the condition "holding other players' strategies fixed". I'll also assume that the strategy space is finite. Given a profile of the other players' ...
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3 votes
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A preference relation is continuous if and if there exists a utility function that represents it

At the heart of Debreu's representation theorem is his so-called "gap theorem": Let $S\subseteq[0,1]$. A gap is a maximal nontrivial interval disjoint from $S$ with an upper and lower bound ...
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2 votes
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On quadratic utility functions

Note that your problem said to assume that $d < 0$. When you divide both sides of an inequality by a negative number, you flip the signs. For example: $$ -2x < 4 \Rightarrow x > -2 $$ Once ...
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2 votes
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Contradicting the von Neumann–Morgenstern axioms

Never tell me the odds Han Solo in Star Wars: The Empire Strikes Back In most domains, people appear to be ambiguity averse (Binmore, Stewart, and Voorhoeve (2012), Ellsberg (1961)), meaning roughly ...
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1 vote
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Find a utility function for a specific problem

Given your requirement there is no unique way of setting the utility as far as I can see. One alternative could be having the following utility: $$U = x^\alpha y^\beta z^\gamma$$ Where $\alpha> \...
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