Don’t mix Risk Management and Competitive Highscores

Games with Risk Management ask players to choose between low variance (or “safe, but low payout”) and high variance (or “risky, but high payout”) strategies.

Single-player games with Competitive Highscores present highscore as the sole goal of the game. Most of these have an online leaderboard.

Game designers combine the two on a continual basis, which I’ve come to believe is a critical mistake. But first, I need to rewind a bit and explain why risk management is a valuable thing to include in games.

Let’s analyze Pass the Pigs, a simple game that happens to beautifully illustrate what this “risk management” thing is all about. Briefly, the rules are as follows:

1) Players take turns rolling adorable pig-shaped dice. One player usually rolls multiple times on his turn.
2) Most rolls score you points.
3) One particular roll, called “Pig Out”, ends your turn and cancels all points you scored that turn.
4) A player rolls until he chooses to stop (at which point he scores the points earned) or gets a Pig Out.
5) First person to 100 points wins.
6) There is actually a different rule where you can lose *all* of your points, but for simplification, let’s pretend it doesn’t exist.​

Your only real decision each turn is to figure out when to stop and take your points.

Perhaps surprisingly, this game is not trivial to play optimally. The dice are unique–they’re pig-shaped–so calculating the probabilities directly is infeasible. Fortunately, we can gather empirical data instead. Amusingly, John C. Kern actually used statistics to analyze Pass the Pigs in the Journal of Statistics Education. Never underestimate science!

In the paper, Kern first rolled the two pigs 6,000 times to get reasonable point estimates on the chance that you’d get each roll. Among other things, he calculated that:

– Your chance of scoring points is 79%, while your chance of a “Pig Out” is about 21%.
– Once you have accumulated 22 points in a turn, rolling again has negative expected value. By that, I mean that that on average, the number of additional points you can win will be outweighed by the 22 points you will lose if you “Pig Out.”
This paper finds that if we roll on 21 and stop at 22, we can expect to go out after about 10 turns.
– If we carry out an “extreme-conservative” strategy of always stopping after one roll, we can expect to “go out” (score 100 points) after 24 turns.
– If we carry out an “extreme-risk” strategy of rolling until we get 100 points, we can expect to go out after 100 turns.​

So the game is solved, now, right? Just roll until you get 22 or more, and then stop every time?

The problem here is that maximizing points is not the actual goal of Pass the Pigs. The goal is to win by going out, and once you factor in your opponent’s score, risk management comes into play. Consider the extreme example of having 0 points versus your opponent’s 99 points. In this situation, your opponent is going to win if he doesn’t Pig Out on his first roll next turn. Your chances of winning are… pretty low. But how do you maximize them?

If you default to the “best” strategy and stop each time you score 22 points, you can expect to go out after 10 turns. Assuming this average run, this means your opponent needs to not pig out 9 times in a row. The chances of this happening are *astronomically* low: less than 0.00001%!

As it turns out, your best strategy in this case is to adopt the normally terrible “extreme-risk” strategy. Your chances of going out immediately are about 1%, which is already several orders of magnitude higher than using the “best” strategy of stopping at 22 points. In fact, extreme-risk’s chance of winning here is slightly higher than 1%; if you fail to go out on the first try, your opponent might whiff his 80% roll once or twice. But either way, the normally awful “extreme-risk” strategy becomes the best option available to you.

This is the core of what risk management is about. Your strategy depends not just on what maximizes points, but your position in points relative to your opponent’s. When you are ahead, you should try to reduce your risk, opting to stop at rolls less than 22. When you’re behind, it’s time to turn on the gambol. In a good risk management game, you often have two equally puzzling challenges: what is the plan that maximizes resources, and what can you afford to risk to carry out this plan?

How does adding players affect optimal strategy? Raising the number of players also increases your inclination to take risks. With each player added, your chance of winning the game goes down, no matter what strategy you use. Similar to how you’re more inclined to bet your wallet if there’s less in it, you should be more inclined to use high risk strategies when your expected chance of winning the game (“equity,” as game theorists say) is low.

If you’re in a game with one other player using the “extreme-risk” strategy, you can easily win almost no matter what you do. Even the terrible “extreme-conservative” strategy will easily beat him, on average. But add in 99 other players doing that, and there’s a pretty good chance you won’t get to see your second turn! At that stage, you need to employ the extreme-risk strategy yourself to even have a chance.

You might think I’ve forgotten about my original thesis, that high-score competition trivializes risk management decisions. But actually, creating a risk-management game where players compete on a global leaderboard functionally increases the number of players to an arbitrarily large number. Just as how the winner of a 100 player game of Pass the Pigs is most likely the one who decided to employ “extreme-risk” and happened to get lucky, if you have risk management in a high-score competition, there is a virtual guarantee that the players on the leaderboard got there by employing an extreme-risk strategy. This, in turn, trivializes the intricacies of risk-management play.

Credit goes to DisquisitorSam for many of the ideas in this post.

Also thanks to Evizaer for looking over this. ^_^