Multi-valued Choice and Extended Preferences

What if there is a tie between two or more alternatives in a voting procedure? One approach is to use a tie-breaking rule to determine the winner. The most popular tie-breaking rule is alphabetical one. In this case, the result of the voting procedure always consists of a single alternative (single-valued choice).

The other approach is to allow ties. If two or more alternatives have the same score in a voting procedure, all such alternatives will be included into the social choice (multi-valued choice). In order to compare all possible sets of alternatives, i.e., all possible social choices, additional assumptions about the preferences of the voters are needed. These assumptions define so-called extended preferences. If a preference of an agent is a linear order over the set of alternatives, an extended preference is a linear order over the set of all possible social choices, i.e. over all subsets of the set of the alternatives.

Extended preferences allow comparing multi-valued choices for an agent. There are 4 ways to construct extended preferences for the case of 3 alternatives, 10 ways for 4 alternatives and 12 ways for 5 alternatives.

Extended Preferences for case of 3 alternatives

There are 4 ways of constructions extended preferences for the case of 3 alternatives. It is assumed that the sincere preference of an agent is: \(a\succ b\succ c\). Underlined are the comparisons, where the way of constructions extended preference differs from other ways.
1) Leximin \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}}\succ\left\{b,c\right\}\succ\left\{c\right\}\] 2) Leximax \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b\right\}}\succ\left\{b,c\right\}\succ\left\{c\right\}\] 3) Risk-averse \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,b,c\right\}\succ\left\{a,c\right\}}\succ\left\{b,c\right\}\succ\left\{c\right\}\] 4) Risk-lover \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,c\right\}\succ\left\{a,b,c\right\}\succ\left\{b\right\}}\succ\left\{b,c\right\}\succ\left\{c\right\}\]

Extended Preferences for the case of 4 alternatives

It is assumed that the sincere preference of an agent is: \(a\succ b\succ c\succ d\).
1) Leximax \[\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{a,b,c\right\}\succ\left\{a,b,c,d\right\}\succ\left\{a,b,d\right\}\succ\left\{a,c\right\}\succ\left\{a,c,d\right\}\succ\left\{a,d\right\}\succ\left\{b\right\}\succ\left\{b,c\right\}\succ\left\{b,c,d\right\}\succ\left\{b,d\right\}\succ\left\{c\right\}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 2) Leximin \[\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}\succ\left\{b,c\right\}\succ\left\{c\right\}\succ\left\{a,d\right\}\succ\left\{a,b,d\right\}\succ\left\{b,d\right\}\succ\left\{a,c,d\right\}\succ\left\{a,b,c,d\right\}\succ\left\{b,c,d\right\}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 3) Average rank, Leximax \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{a,b,c,d\right\}\succ\underline{\left\{a,d\right\}\succ\left\{b,c\right\}}}\succ\left\{a,c,d\right\}\succ\underline{\left\{b,c,d\right\}\succ\left\{b,d\right\}\succ\left\{c\right\}}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 4) Average rank, Leximin \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}}\succ\left\{a,b,d\right\}\succ\underline{\underline{\left\{b,c\right\}\succ\left\{a,d\right\}}\succ\left\{a,b,c,d\right\}}\succ\left\{a,c,d\right\}\succ\underline{\left\{c\right\}\succ\left\{b,d\right\}\succ\left\{b,c,d\right\}}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 5) Average rank, Risk-averse \[\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{b\right\}\succ\left\{a,b,c\right\}\succ\left\{a,с\right\}\succ\left\{a,b,d\right\}\succ\left\{b,c\right\}\succ\left\{a,b,c,d\right\}\succ\left\{a,d\right\}\succ\left\{a,c,d\right\}\succ\left\{c\right\}\succ\left\{b,c,d\right\}\succ\left\{b,d\right\}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 6) Average rank, Risk-lover \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,c\right\}\succ\left\{a,b,c\right\}\succ\left\{b\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{a,d\right\}\succ\left\{a,b,c,d\right\}\succ\left\{b,c\right\}}\succ\left\{a,c,d\right\}\succ\underline{\left\{b,d\right\}\succ\left\{b,c,d\right\}\succ\left\{c\right\}}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 7) Average rank, decreasing cardinality, risk-averse \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{a,b,c,d\right\}\succ\underline{\left\{b,c\right\}\succ\left\{a,d\right\}}}\succ\left\{a,c,d\right\}\succ\underline{\left\{b,c,d\right\}\succ\left\{b,d\right\}\succ\left\{c\right\}}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 8) Average rank, increasing cardinality, risk-lover \[\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}}\succ\left\{a,b,d\right\}\succ\underline{\underline{\left\{a,d\right\}\succ\left\{b,c\right\}}\succ\left\{a,b,c,d\right\}}\succ\left\{a,c,d\right\}\succ\underline{\left\{c\right\}\succ\left\{b,d\right\}\succ\left\{b,c,d\right\}}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 9) Probability of the worst \[\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{a,c\right\}\succ\left\{a,d\right\}\succ\left\{a,b,с\right\}\succ\left\{a,b,d\right\}\succ\left\{a,c,d\right\}\succ\left\{a,b,c,d\right\}\succ\left\{b\right\}\succ\left\{b,c\right\}\succ\left\{b,d\right\}\succ\left\{b,c,d\right\}\succ\left\{c\right\}\succ\left\{c,d\right\}\succ\left\{d\right\}\] 10) Probability of the best \[\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{b\right\}\succ\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b,c\right\}\succ\left\{c\right\}\succ\left\{a,b,c,d\right\}\succ\left\{a,b,d\right\}\succ\left\{a,c,d\right\}\succ\left\{b,c,d\right\}\succ\left\{a,d\right\}\succ\left\{b,d\right\}\succ\left\{c,d\right\}\succ\left\{d\right\}\]

Extended Preferences for the case of 5 alternatives

It is assumed that the sincere preference of an agent is: \(a\succ b\succ c\succ d\succ e\).
1) Leximin \[\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}\succ\left\{b,c\right\}\succ\left\{c\right\}\succ\left\{a,d\right\}\succ\left\{a,b,d\right\}\succ\left\{b,d\right\}\succ\left\{a,c,d\right\}\succ\left\{a,b,c,d\right\}\succ\left\{b,c,d\right\}\succ\left\{c,d\right\}\succ\left\{d\right\}\succ\left\{a,e\right\}\succ\left\{a,b,e\right\}\succ\left\{b,e\right\}\succ\left\{a,c,e\right\}\succ\left\{a,b,c,e\right\}\succ\left\{b,c,e\right\}\succ\left\{c,e\right\}\succ\left\{a,d,e\right\}\succ\left\{a,b,d,e\right\}\succ\left\{b,d,e\right\}\succ\left\{a,с,d,e\right\}\succ\left\{a,b,c,d,e\right\}\succ\left\{b,c,d,e\right\}\succ\left\{c,d,e\right\}\succ\left\{d,e\right\}\succ\left\{e\right\}\] 2) Leximax \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{a,b,c\right\}\succ\left\{a,b,c,d\right\}\succ\left\{a,b,c,d,e\right\}\succ\left\{a,b,c,e\right\}\succ\left\{a,b,d\right\}\succ\left\{a,b,d,e\right\}\succ\left\{a,b,e\right\}\succ\left\{a,c\right\}\succ\left\{a,c,d\right\}\succ\left\{a,c,d,e\right\}\succ\left\{a,c,e\right\}\succ\left\{a,d\right\}\succ\left\{a,d,e\right\}\succ\left\{a,e\right\}\succ\\\succ\left\{b\right\}\succ\left\{b,c\right\}\succ\left\{b,c,d\right\}\succ\left\{b,c,d,e\right\}\succ\left\{b,c,e\right\}\succ\left\{b,d\right\}\succ\left\{b,d,e\right\}\succ\left\{b,e\right\}\succ\left\{c\right\}\succ\left\{c,d\right\}\succ\left\{c,d,e\right\}\succ\left\{c,e\right\}\succ\left\{d\right\}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 3) Average rank Leximin \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{b,c\right\}\succ\left\{a,d\right\}\succ\left\{a,b,c,d\right\}}\succ\underline{\left\{a,c,d\right\}\succ\left\{a,b,e\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{c\right\}\succ\left\{b,d\right\}\succ\left\{b,c,d\right\}\succ\left\{a,e\right\}\succ\left\{a,c,e\right\}\succ\left\{a,b,d,e\right\}\succ\left\{a,b,c,d,e\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{b,c,e\right\}\succ\left\{a,d,e\right\}}\succ\underline{\left\{c,d\right\}\succ\left\{b,e\right\}\succ\left\{b,c,d,e\right\}}\succ\left\{b,d,e\right\}\succ\underline{\left\{d\right\}\succ\left\{c,e\right\}\succ\left\{c,d,e\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 4) Average rank Leximax \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{a,b,c,d\right\}\succ\left\{a,d\right\}\succ\left\{b,c\right\}}\succ\underline{\left\{a,b,e\right\}\succ\left\{a,c,d\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{a,b,c,d,e\right\}\succ\left\{a,b,d,e\right\}\succ\left\{a,c,e\right\}\succ\left\{a,e\right\}\succ\left\{b,c,d\right\}\succ\left\{b,d\right\}\succ\left\{c\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{a,d,e\right\}\succ\left\{b,c,e\right\}}\succ\underline{\left\{b,c,d,e\right\}\succ\left\{b,e\right\}\succ\left\{c,d\right\}}\succ\left\{b,d,e\right\}\succ\underline{\left\{c,d,e\right\}\succ\left\{c,e\right\}\succ\left\{d\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 5) Average rank, risk-averse \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,b,c\right\}\succ\left\{a,c\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{b,c\right\}\succ\left\{a,b,c,d\right\}\succ\left\{a,d\right\}}\succ\underline{\left\{a,c,d\right\}\succ\left\{a,b,e\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{c\right\}\succ\left\{b,c,d\right\}\succ\left\{b,d\right\}\succ\left\{a,b,c,d,e\right\}\succ\left\{a,b,d,e\right\}\succ\left\{a,c,e\right\}\succ\left\{a,e\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{b,c,e\right\}\succ\left\{a,d,e\right\}}\succ\underline{\left\{c,d\right\}\succ\left\{b,c,d,e\right\}\succ\left\{b,e\right\}}\succ\left\{b,d,e\right\}\succ\underline{\left\{d\right\}\succ\left\{c,d,e\right\}\succ\left\{c,e\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 6) Average rank, risk-lover \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,c\right\}\succ\left\{a,b,c\right\}\succ\left\{b\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{a,d\right\}\succ\left\{a,b,c,d\right\}\succ\left\{b,c\right\}}\succ\underline{\left\{a,b,e\right\}\succ\left\{a,c,d\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{a,e\right\}\succ\left\{a,c,e\right\}\succ\left\{a,b,d,e\right\}\succ\left\{a,b,c,d,e\right\}\succ\left\{b,d\right\}\succ\left\{b,c,d\right\}\succ\left\{c\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{a,d,e\right\}\succ\left\{b,c,e\right\}}\succ\underline{\left\{b,e\right\}\succ\left\{b,c,d,e\right\}\succ\left\{c,d\right\}}\succ\left\{b,d,e\right\}\succ\underline{\left\{c,e\right\}\succ\left\{c,d,e\right\}\succ\left\{d\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 7) Average rank, decreasing cardinality, risk-lover \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{a,b,c,d\right\}\succ\underline{\left\{a,d\right\}\succ\left\{b,c\right\}}}\succ\underline{\left\{a,b,e\right\}\succ\left\{a,c,d\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{a,b,c,d,e\right\}\succ\left\{a,b,d,e\right\}\succ\underline{\left\{a,c,e\right\}\succ\left\{b,c,d\right\}}\succ\underline{\left\{a,e\right\}\succ\left\{b,d\right\}}\succ\left\{c\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{a,d,e\right\}\succ\left\{b,c,e\right\}}\succ\underline{\left\{b,c,d,e\right\}\succ\underline{\left\{b,e\right\}\succ\left\{c,d\right\}}}\succ\left\{b,d,e\right\}\succ\underline{\left\{c,d,e\right\}\succ\left\{c,e\right\}\succ\left\{d\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 8) Average rank, decreasing cardinality, risk-averse \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b\right\}}\succ\left\{a,b,d\right\}\succ\underline{\left\{a,b,c,d\right\}\succ\underline{\left\{b,c\right\}\succ\left\{a,d\right\}}}\succ\underline{\left\{a,c,d\right\}\succ\left\{a,b,e\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{a,b,c,d,e\right\}\succ\left\{a,b,d,e\right\}\succ\underline{\left\{b,c,d\right\}\succ\left\{a,c,e\right\}}\succ\underline{\left\{b,d\right\}\succ\left\{a,e\right\}}\succ\left\{c\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{b,c,e\right\}\succ\left\{a,d,e\right\}}\succ\underline{\underline{\left\{c,d\right\}\succ\left\{b,e\right\}}\succ\left\{b,c,d,e\right\}}\succ\left\{b,d,e\right\}\succ\underline{\left\{c,d,e\right\}\succ\left\{c,e\right\}\succ\left\{d\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 9) Average rank, increasing cardinality, risk-lover \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}}\succ\left\{a,b,d\right\}\succ\underline{\underline{\left\{a,d\right\}\succ\left\{b,c\right\}}\succ\left\{a,b,c,d\right\}}\succ\underline{\left\{a,b,e\right\}\succ\left\{a,c,d\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{c\right\}\succ\underline{\left\{a,e\right\}\succ\left\{b,d\right\}}\succ\underline{\left\{a,c,e\right\}\succ\left\{b,c,d\right\}}\succ\left\{a,b,d,e\right\}\succ\left\{a,b,c,d,e\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{a,d,e\right\}\succ\left\{b,c,e\right\}}\succ\underline{\underline{\left\{b,e\right\}\succ\left\{c,d\right\}}\succ\left\{b,c,d,e\right\}}\succ\left\{b,d,e\right\}\succ\underline{\left\{d\right\}\succ\left\{c,e\right\}\succ\left\{c,d,e\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 10) Average rank, decreasing cardinality, risk-averse \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\underline{\left\{b\right\}\succ\left\{a,c\right\}\succ\left\{a,b,c\right\}}\succ\left\{a,b,d\right\}\succ\underline{\underline{\left\{b,c\right\}\succ\left\{a,d\right\}}\succ\left\{a,b,c,d\right\}}\succ\underline{\left\{a,c,d\right\}\succ\left\{a,b,e\right\}}\succ\left\{a,b,c,e\right\}\succ\\\succ\underline{\left\{c\right\}\succ\underline{\left\{b,d\right\}\succ\left\{a,e\right\}}\succ\underline{\left\{b,c,d\right\}\succ\left\{a,c,e\right\}}\succ\left\{a,b,d,e\right\}\succ\left\{a,b,c,d,e\right\}}\succ\\\succ\left\{a,c,d,e\right\}\succ\underline{\left\{b,c,e\right\}\succ\left\{a,d,e\right\}}\succ\underline{\underline{\left\{c,d\right\}\succ\left\{b,e\right\}}\succ\left\{b,c,d,e\right\}}\succ\left\{b,d,e\right\}\succ\underline{\left\{d\right\}\succ\left\{c,e\right\}\succ\left\{c,d,e\right\}}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\] 11) Probability of the best \[\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{a,c\right\}\succ\left\{a,d\right\}\succ\left\{a,e\right\}\succ\left\{a,b,c\right\}\succ\left\{a,b,d\right\}\succ\left\{a,b,e\right\}\succ\left\{a,c,d\right\}\succ\left\{a,c,e\right\}\succ\left\{a,d,e\right\}\succ\left\{a,b,c,d\right\}\succ\left\{a,b,c,e\right\}\succ\left\{a,b,d,e\right\}\succ\left\{a,c,d,e\right\}\succ\left\{a,b,c,d,e\right\}\succ\left\{b\right\}\succ\left\{b,c\right\}\succ\left\{b,d\right\}\succ\left\{b,e\right\}\succ\left\{b,c,d\right\}\succ\left\{b,c,e\right\}\succ\left\{b,d,e\right\}\succ\left\{b,c,d,e\right\}\succ\left\{c\right\}\succ\left\{с,d\right\}\succ\left\{c,e\right\}\succ\left\{c,d,e\right\}\succ\left\{d\right\}\succ\left\{d,e\right\}\succ\left\{e\right\}\] 12) Probability of the worst \[\begin{matrix}\left\{a\right\}\succ\left\{a,b\right\}\succ\left\{b\right\}\succ\left\{a,b,c\right\}\succ\left\{a,c\right\}\succ\left\{b,c\right\}\succ\left\{c\right\}\succ\left\{a,b,c,d\right\}\succ\left\{a,b,d\right\}\succ\left\{a,c,d\right\}\succ\left\{b,c,d\right\}\succ\left\{a,d\right\}\succ\left\{b,d\right\}\succ\left\{c,d\right\}\succ\left\{d\right\}\succ\\\succ\left\{a,b,c,d,e\right\}\succ\left\{a,b,c,e\right\}\succ\left\{a,b,d,e\right\}\succ\left\{a,c,d,e\right\}\succ\left\{b,c,d,e\right\}\succ\left\{a,b,e\right\}\succ\left\{a,c,e\right\}\succ\left\{b,c,e\right\}\succ\left\{a,d,c\right\}\succ\left\{b,d,e\right\}\succ\left\{c,d,e\right\}\succ\left\{a,e\right\}\succ\left\{b,c\right\}\succ\left\{c,e\right\}\succ\left\{d,e\right\}\succ\left\{e\right\}\\\end{matrix}\]

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