On the Trail of Lost Pennies

We introduce a two-person non-zero-sum random-turn game that is a variant of the stake-governed games introduced recently in [HP2022]. We call the new game the Trail of Lost Pennies. At time zero, a counter is placed at a given integer location: $X_0 = k \in \mathbb{Z}$, say. At the $i$-th turn (for $i \in \mathbb{N}_+$), Maxine and Mina place non-negative stakes, $a_i$ and $b_i$, for which each pays from her own savings. Maxine is declared to be the turn victor with probability $\tfrac{a_i}{a_i+b_i}$; otherwise, Mina is. If Maxine wins the turn, she will move the counter one place to the right, so that $X_i = X_{i-1} +1$; if Mina does so, the counter will move one place to the left, so that $X_i = X_{i-1} -1$. If $\liminf X_i = \infty$, then Maxine wins the game; if $\limsup X_i = -\infty$, then Mina does. (A special rule is needed to treat the remaining, indeterminate, case.) When Maxine wins, she receives a terminal payment of $m_\infty$, while Mina receives $n_\infty$. If Mina wins, these respective receipts are $m_{-\infty}$ and $n_{-\infty}$. The four terminal payment values are supposed to be real numbers that satisfy $m_\infty > m_{-\infty}$ and $n_\infty < n_{-\infty}$, where these bounds accord with the notion that Maxine wins when the counter ends far to the right, and that Mina does so when it reaches far to the left. Each player is motivated to offer stakes at each turn of the game, in order to secure the higher terminal payment that will arise from her victory; but since these stake amounts accumulate to act as a cost depleting the profit arising from victory, each player must also seek to control these expenses. In this article, we study the Trail of Lost Pennies, formulating strategies for the two players and defining and analysing Nash equilibria in the game.