Hold or roll: reaching the goal in jeopardy race games

Abstract

We consider a class of dynamic tournaments in which two contestants are faced with a choice between two courses of action. The first is a riskless option (“hold”) of maintaining the resources the contestant already has accumulated in her turn and ceding the initiative to her rival. The second is the bolder option (“roll”) of taking the initiative of accumulating additional resources, and thereby moving ahead of her rival, while at the same time sustaining a risk of temporary setback. We study this tournament in the context of a jeopardy race game (JRG), extend the JRG to $N > 2$ contestants, and construct its equilibrium solution. Compared to the equilibrium solution, the results of three experiments reveal a dysfunctional bias in favor of the riskless option. This bias is substantially mitigated when the contestants are required to commit in advance how long to pursue the risky course of action.

Appendices

Appendix 1

1.1 Numerical solution of the equilibrium strategy for the 3-person JRG: the decision method

We are looking at each state from the viewpoint of each of the three players to obtain the values of$P_\mathrm{A} ,P_\mathrm{B} ,P_\mathrm{C}$. Certain of these probabilities must be the same. Namely, at any stage the only thing that distinguishes one player from another is whether they control the die, or are the next in line to control the die or are the second in line. Therefore, $(i,j,k,t,\mathrm{A}),(k,i,j,t,\mathrm{B}),(j,k,i,t,\mathrm{C})$ are 3 states that can all be described the same way as: the player in control of the die has $i$ points and $t$ so far this turn while the next player has $j$ points and the player after that has $k$ points. Therefore,

$$\begin{aligned} P_\mathrm{A} (i,j,k,t,\mathrm{A})&= P_\mathrm{B} (k,i,j,t,\mathrm{B})=P_\mathrm{C} (j,k,i,t,\mathrm{C}) \nonumber \\ P_\mathrm{B} (i,j,k,t,\mathrm{A})&= P_\mathrm{C} (k,i,j,t,\mathrm{B})=P_\mathrm{A} (j,k,i,t,\mathrm{C}) \\ P_\mathrm{C} (i,j,k,t,\mathrm{A})&= P_\mathrm{A} (k,i,j,t,B)=P_\mathrm{B} (j,k,i,t,\mathrm{C}). \nonumber \end{aligned}$$

(12)

Note that (12) is not of the same structure as (2), which relates the three probabilities for a specific state s. Because of (2), we never need to explicitly compute $P_\mathrm{C}$ for any state. We can also avoid all computations involving C when he is the player in control of the die by using (12). That is, $P_\mathrm{A} (j,k,i,t,\mathrm{C})$ and $P_\mathrm{B} (j,k,i,t,\mathrm{C})$ can be replaced using

$$\begin{aligned} \begin{aligned} P_\mathrm{A} (j,k,i,t,\mathrm{C})&= P_\mathrm{B} (i,j,k,t,\mathrm{A}) \\ P_\mathrm{B} (j,k,i,t,\mathrm{C})&= P_\mathrm{A} (k,i,j,t,\mathrm{B}). \end{aligned} \end{aligned}$$

(13)

Therefore, we have reduced the problem to recursive computations for players A and B. In fact from (7) we can also eliminate $P_\mathrm{B} (-,-,-,-,\mathrm{B})$ and use only storage space to hold $P_\mathrm{A} (-,-,-,-,\mathrm{A}), \quad P_\mathrm{A} (-,-,-,-,\mathrm{B}), \quad P_\mathrm{B} (-,-,-,-,\mathrm{A}).$ This requires three matrices of size $M^{4}$. (With $M=60$, that is almost 13 million entries per matrix.)

We now specify the initial conditions and boundary conditions. Assume that all matrices are first set to 0. In the following, assume $i,j,k$ are all less than $M$ unless otherwise stated. The boundary conditions are

• If $i+t\ge M$ then $P_\mathrm{A} (i,j,k,t,\mathrm{A})=1$.

• If $i\ge M$ then $P_\mathrm{A} (i,j,k,t,\mathrm{B})=1$.

• If $j\ge M$ then $P_\mathrm{B} (i,j,k,t,\mathrm{A})=1$.

• If $i+t\ge M$ then $P_\mathrm{A} (i,j,k,t,\mathrm{A})=1$.

Now, we start at state $s=(i,j,k,t,\mathrm{A})$ and write out (3) and (4)

$$\begin{aligned} P_\mathrm{A} (s)&= \max \left\{ P_\mathrm{A} (i+t,j,k,0,\mathrm{B}),\left[ P_\mathrm{A} (i,j,k,0,\mathrm{B})\right. \right. \nonumber \\&\left. \left. +\sum _{m=2}^6 {P_\mathrm{A} (i,j,k,t+m,\mathrm{A})}\right] \Big /6\right\} . \end{aligned}$$

(14)

Note: (14) turns out to be the same equations as $({4}^{\prime })$. This fills up the $P_\mathrm{A} (-,-,-{,}-,\mathrm{A})$ matrix using $P_\mathrm{A} (-,-,-,-,\mathrm{A})$ and $P_\mathrm{A} (-,-,-,-,\mathrm{B})$ matrices. With the same state $s$ we now use (5)

$$\begin{aligned} P_\mathrm{B} (s)= \left\{ \begin{array}{l@{\quad }l} P_\mathrm{B} (i+t,j,k,0,\mathrm{B}), &{} \hbox {if } \mathrm{A} \hbox { holds at } s \\ {[}P_\mathrm{B} (i,j,k,0,\mathrm{B})+\sum \nolimits _{m=2}^6 P_\mathrm{B} (i,j,k,t+m,\mathrm{A}){]}/6,&{} \hbox {if} \mathrm{A} \hbox { rolls at } s. \\ \end{array} \right. \end{aligned}$$

Rewrite this using (7)

$$\begin{aligned} P_\mathrm{B} (s)=\left\{ \begin{array}{l@{\quad }l} P_\mathrm{A} (j,k,i+t,0,\mathrm{A}), &{} \hbox {if } \mathrm{A} \hbox { holds at } s \\ {[}P_\mathrm{A} (j,k,i,0,\mathrm{A})&{}\\ \quad +\sum \nolimits _{m=2}^6 {P_\mathrm{B} (i,j,k,t+m,\mathrm{A})} {]}/6, &{} \hbox {if } \mathrm{A} \hbox { rolls at } s. \\ \end{array} \right. \end{aligned}$$

(15)

This fills up the $P_\mathrm{B} (-,-,-,-,\mathrm{A})$ matrix using $P_\mathrm{A} (-,-,-,-,\mathrm{A})$ and $P_\mathrm{B} (-,-,-,-,\mathrm{A})$. Now consider (6) for the same state $s=(i,j,k,t,\mathrm{A})$

$$\begin{aligned} P_\mathrm{C} (s)=\left\{ \begin{array}{l@{\quad }l} P_\mathrm{C} (i+t,j,k,0,\mathrm{B}), &{} \hbox {if } \mathrm{A} \hbox { holds at } s \\ {[}P_\mathrm{C} (i,j,k,0,\mathrm{B})+\sum \nolimits _{m=2}^6 {P_\mathrm{C} (i,j,k,t+m,\mathrm{A})} {]}/6, &{} \hbox {if } \mathrm{A} \hbox { rolls at } s. \\ \end{array} \right. \end{aligned}$$

We may rewrite (7) as

$$\begin{aligned} P_\mathrm{A} (k,i,j,t,\mathrm{B})=\left\{ \begin{array}{l@{\quad }l} P_\mathrm{B} (j,k,i+t,0,\mathrm{A}), &{} \hbox {if }\mathrm{A} \hbox { holds at }s \\ {[}P_\mathrm{B} (j,k,i,0,\mathrm{A})&{}\\ \quad +\sum \nolimits _{m=2}^6 {P_\mathrm{A} (k,i,j,t+m,\mathrm{B})}{]}/6, &{} \hbox {if } \mathrm{A} \hbox { rolls at }s.\\ \end{array} \right. \end{aligned}$$

(16)

This fills up the $P_\mathrm{A} (-,-,-,-,\mathrm{B})$ matrix using $P_\mathrm{A} (-,-,-,-,\mathrm{B})$ and $P_\mathrm{B} (-,-,-,-,\mathrm{A})$. Note that in (16) we have to check whether A holds or rolls at $s=(i,j,k,t,\mathrm{A})$ and then update $P_\mathrm{A} (k,i,j,t,\mathrm{B})$, not $P_\mathrm{A} (i,j,k,t,\mathrm{B})$.

We now compute (14), (15), and (16). We need to set up space for the matrices $P_\mathrm{A} (-,-,-,-,\mathrm{A}), P_\mathrm{A} (-,-,-,-,\mathrm{B}), P_\mathrm{B} (-,-,-,-,\mathrm{A})$ and a matrix of the same size to store the (current) strategy being used by player A. We can use “asynchronous updating” in this dynamic programming problem. This means the same three matrices appear in (14), (15), and (16) whether they are on the right or left side of the equations. This speeds up convergence and eliminates the need for a partitioning of the states as described in Neller and Presser (2004, p. 31). The probabilities were always increasing over the iterations so the terminating condition was chosen as making Eq. (2) for the starting state as close to 1 as desired: this required 25–40 iterations for 0.9999.

Convergence was fast for small $M$ values from a few seconds up to 90 s for $M=30$ to an hour for $M=60$. The algorithm was programmed using MATLAB.

Appendix 2

1.1 Numerical solution for the equilibrium strategy of the 2-player JRG: the strategy method

On each turn, assume a finite maximum number of dice can be rolled from $d$ to $d_{\max }.$ ¹ Using notation similar to Neller and Presser (2005), let $\pi (d, t)$ represent the probability that rolling $d$ dice $(0< d < d_{\max })$ results in a turn score of $t \ge 0.$ Then,

$$\begin{aligned} \pi (d, t)=\left\{ \begin{array}{l@{\quad }l} 1/6, &{} d=1\;\mathrm{and}\;t\in \left\{ {0, 2,3,4,5,6} \right\} ;\\ 0, &{} d=1\;\mathrm{and}\;t\notin \left\{ {0,2,3,4,5,6} \right\} ;\\ \pi (d-1, 0)&{}\\ \quad +1/6 \sum \nolimits _{t=2}^{6(d-1)} \pi (d-1,t), &{} d>1\;\mathrm{and}\; t=0;\\ 1/6 \sum \nolimits _{r=2}^{\min (6,t-2)} \pi (d-1,t-r), &{} \mathrm{otherwise}. \\ \end{array} \right. \end{aligned}$$

(17)

Under the strategy method, the player is not deciding between roll or hold, rather, how many dice to roll during her turn. She chooses the number of dice $d$ that maximizes here probability of winning. Let $P_{i,j}$ represent the probability that the player wins at state $i, j$. As in the standard JRG, if $i \ge 100,$ then $P_{i,j} = 1$ as the player has sufficient points to win the game. In other states where neither player has sufficient points to win the game $(0 \le i, j < 100),$ the probability that the player wins is given by

$$\begin{aligned} P_{i, j} = \mathop {\max }\limits _{0<d\le d_{\max }} \sum \limits _{t=0}^{6d} \pi (d,t)(1-P_{j,i+t}). \end{aligned}$$

(18)

Neller and Presser point out that the shape (or surface) of the equilibrium solution for the strategy method (when depicted as a three-dimensional plot on states $i,j$ and dice d) is similar to that of the standard JRG plotted on states $i, j, t$. If one was to multiply the equilibrium number of dice to roll by 4 (which yields the expected value of a successful turn), then this product approximates the roll/hold threshold values for the standard JRG.