I need to generically define a strict preference profile with Condorcet cicles when the number of players and alternatives coincide. To illustrate my problem, consider the following four-player & four-alternative example.

Let $A=\{a_1,a_2,a_3,a_4\}$ be a set of four alternatives, let $N=\{1,2,3,4\}$ be a set of four players and let $P=(P_i)_{i\in N}$ be a preference profile (i.e., linear order profile) on alternatives such that \begin{gather} a_1P_1a_2P_1a_3P_1a_4\\ a_2P_2a_3P_2a_4P_2a_1\\ a_3P_3a_4P_3a_1P_3a_2\\ a_4P_4a_1P_4a_2P_4a_3 \end{gather}

In the preference profile above, the top-ranked alternative of player $i$ becomes the bottom-ranked alternative of player $i+1$, and so on.

In order to generalise the strict preference profile above to an arbitrary (but equal) number of players and alternatives, I had thought of writing the following: let $P=(P_i)_{i\in N}$ satisfy, for every player $i\in N$, \begin{gather} a_iP_ia_{i+1}P_i\dots P_ia_{n+i-1} \end{gather} Then, for player $i=1$, we obtain $a_i=a_1$, $a_{i+1}=a_2$, $a_{i+2}=a_3$ and $a_{n+1-i}=a_{4+1-1}=a_4$; and thus, \begin{gather} a_1P_1a_2P_1a_3P_1a_4 \end{gather} Hence, for player $i=1$, this approach works.

However, for player $i=2$, we obtain $a_i=a_2$, $a_{i+1}=a_3$, $a_{i+2}=a_4$ and $a_{n+1-i}=a_{4+1-2}=a_3$; and thus, \begin{gather} a_2P_2a_3P_2a_4P_2a_3 \end{gather} Hence, for player $i=2$ (and by a similar token, for all players $i\neq 1$), my approach does not work.

Hence, how to formally define the preference profile in which the top alternative of one player becomes the bottom alternative of the next player?

I need to generically define a strict preference profile with Condorcet cicles when the number of players and alternatives coincide. To illustrate my problem, consider the following four-player & four-alternative example.

Let $A=\{a_1,a_2,a_3,a_4\}$ be a set of four alternatives, let $N=\{1,2,3,4\}$ be a set of four players and let $P=(P_i)_{i\in N}$ be a preference profile (i.e., linear order profile) on alternatives such that \begin{gather} a_1P_1a_2P_1a_3P_1a_4\\ a_2P_2a_3P_2a_4P_2a_1\\ a_3P_3a_4P_3a_1P_3a_2\\ a_4P_4a_1P_4a_2P_4a_3 \end{gather}

In the preference profile above, the top-ranked alternative of player $i$ becomes the bottom-ranked alternative of player $i+1$, and so on.

In order to generalise the strict preference profile above to an arbitrary (but equal) number of players and alternatives, I had thought of writing the following: let $P=(P_i)_{i\in N}$ satisfy, for every player $i\in N$, \begin{gather} a_iP_ia_{i+1}P_i\dots P_ia_{n+i-1} \end{gather} Then, for player $i=1$, we obtain $a_i=a_1$, $a_{i+1}=a_2$, $a_{i+2}=a_3$ and $a_{n+1-i}=a_{4+1-1}=a_4$; and thus, \begin{gather} a_1P_1a_2P_1a_3P_1a_4 \end{gather} Hence, for player $i=1$, this approach works.

However, for player $i=2$, we obtain $a_i=a_2$, $a_{i+1}=a_3$, $a_{i+2}=a_4$ and $a_{n+1-i}=a_{4+1-2}=a_3$; and thus, \begin{gather} a_2P_2a_3P_2a_4P_2a_3 \end{gather} Hence, for player $i=2$ (and by a similar token, for all players $i\neq 1$), my approach does not work.

Hence, how to formally define the preference profile in which the top alternative of one player becomes the bottom alternative of the next player?

EDIT A friend of mine suggested using modular arithmetic to do what I need to do; and while I think he’s right, I don’t see how to proceed.