# A Giant Mistake

A couple weeks ago Jerry Holkins introduced the "C-Team" audience to a game called Giants and Halflings. At first glance, the game looks to be a little unfair for the Giants. However, after some investigation I can confidently say this game is a complicated way for a giant to lose his stuff. In this post I'll explore precisely how bad an idea it is for a giant to battle two halflings.

**Rules**

Giants and Halflings is an asymmetrical dice game, with one player (the giant) rolling a D10 and the other (the halflings) rolling two D6's. Thematically, the point is for the halflings to hit the giant in the knee without getting stomped or jumping into his mouth.

A game starts with a roll of the giant. If the die returns one, then the giant stomps the halflings, and the halflings lose. All other D10 rolls are combative.

The halflings' response is the sum of their two dice, with three possible results. If the dice add to two, the game is a draw, even in response to a stomp. If the dice add to less than the giant's roll, or above ten, then the halflings lose. And, a roll that adds to the giant's roll or higher, without exceeding ten, is a halfling victory.

Finally the payout. in a friendly game every bet returns 1:1. However, less than friendly games are less than simple. On a draw, the giant and the halflings keep their bets. When the giant wins, he keeps the halflings' bet. And, the returns for a halfling victory are based on the giant's roll, and best shown with an itemized list.

- If the giant rolls a two, or three, then the halflings take giant's bet.
- If the giant rolls a four or five, then the halflings take twice the giant's bet.
- If the giant rolls a six or seven, then the halflings take three times the giant's bet.
- If the giant rolls an eight or nine, then the halflings take four times the giant's bet.
- If the giant rolls a ten, then the halflings take five times the giant's bet.

Below I've created a table showing the giant's profits and losses based on dice rolls. $$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} \text{Halfling} & \multicolumn{2}{c}{\text{Giant}} \\ \text{(Odds)} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline 2 (1) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 (2) & 1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 4 (3) & 1 & -1 & -1 & -2 & 1 & 1 & 1 & 1 & 1 & 1 \\ 5 (4) & 1 & -1 & -1 & -2 & -2 & 1 & 1 & 1 & 1 & 1 \\ 6 (5) & 1 & -1 & -1 & -2 & -2 & -3 & 1 & 1 & 1 & 1 \\ 7 (6) & 1 & -1 & -1 & -2 & -2 & -3 & -3 & 1 & 1 & 1 \\ 8 (5) & 1 & -1 & -1 & -2 & -2 & -3 & -3 & -4 & 1 & 1 \\ 9 (4) & 1 & -1 & -1 & -2 & -2 & -3 & -3 & -4 & -4 & 1\\ 10 (3) & 1 & -1 & -1 & -2 & -2 & -3 & -3 & -4 & -4 & -5 \\ 11 (2) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 12 (1) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array} $$

**Single Game Odds of Payout**

On the face of it, the payout table above shows 110 conditions and their impact on the giant's coffers. But most of the halflings' possible values can be made with more than one unique roll of the D6 dice. As such, there are actually 360 unique rolls.

A deeper analysis of the possible outcomes of a single game perpetuates the idea that giants shouldn't play this game. Of the 360 possibilities there are 166 wins for the giant (46.1%), 10 draws (2.8%), and 184 losses (51.1%). The giant's odds aren't horrible, but they're not as good as a coin toss.

What's worse, a loss for the giant can result in him losing more than the halfling's bet. Of the 184 possible losses, 120 pay the halfling more than 1:1.

**Win Ratio and Payout**

To make money at gambling, in the long run, the giant needs to win more money than he loses. That is, the amount he receives, from wins, needs to exceed what he pays for losses. The easiest way to do this is to compare the giant's win:loss ratio with the payout ratio. As an example, if the giant pays 4:1 for a loss, then he needs to win more than four games for each loss to make a profit.

Since the payouts vary for each D10 roll, I'll look at the win:loss ratio for each payout. For the sake of simplicity, I've put the data in a table: $$ \begin{array}{c|c|c} \begin{array}{c} \text{House} \\ \text{Roll} \end{array} & \begin{array}{c} \text{Win} \\ \text{Ratio} \end{array} & \text{Payout} \\ \hline 1,2,\text{ or }3 & 41:64=0.67:1 & 1:1 \\ 4\text{ or }5 & 13:57=0.48:1 & 2:1 \\ 6\text{ or }7 & 29:41=0.66:1 & 3:1 \\ 8\text{ or }9 & 51:19=2.68:1 & 4:1 \\ 10 & 32:3=10.67:1 & 5:1 \\ \text{Total} & 166:184=0.90:1 & 337:166=2.03:1 \end{array} $$ The only D10 roll that stands a chance of profit is 10, and on the whole the giant will very probably lose his shirt and his pants.

**Odds of Profit Over Multiple Games**

In any one game the giant isn't very likely to win. But, how likely is it that the giant will make any profits over several games? Finding profits requires that the giant win more than he gives to the halflings. Below I've put together a graph showing the percent probability that the giant will profit after some number of games.

The curve shows rapid decay in probability that asymptotically approaches not a chance in hell.

What's most interesting about this graph is the hitch between the second and third games. This is because game two profits depend on all but one game being a draw much more than any other, and the odds of a draw are very low. By the time the third game rolls around, the flagging odds are buoyed by the higher probability that the giant will lose at 1:1.

I don't see any reason why someone would play this game as the giant, but at least it provided some fun analysis.