Bene Factum

2013/02/08

Taking your shot

Filed under: Gaming Blog — Tags: , , — AlexWeldon @ 8:10 pm

I’ve been playing Sid Sackson’s seminal classic Can’t Stop the last few days, in its new iPad form. Although I’d (shamefully) never played it until now, simply hearing it described was enough for it to serve as partial inspiration for my own Picnic Blitz, as some reviewers have noticed.

Although I found the AI a challenge for my first few attempts, I learned quickly and, as is often the case with board game AIs, was soon able to defeat it a large majority of the time in one-on-one games. Its biggest weakness, I’ve observed, is that it does not to give enough (or perhaps any, it is hard to tell) consideration to the likelihood that you will be able to win on your next move. It will make an otherwise-sensible preparatory move to improve its odds of completing a column on its next move, without realizing that it isn’t likely to get a next move, and should instead shoot for a win immediately, even if the odds of success are small.

The choice between attempting to win on one’s current move or instead building up power to try to win on a subsequent move is a common dilemma in games; it’s embodied in a very pure form in Can’t Stop (and other press-your-luck dice games such as Nada), but it occurs frequently in other games in a more complex, harder-to-quantify way; deciding when to stop building units and launch a final assault in a military game, or whether to call an opponent’s all-in in poker vs. folding and trying to find a better spot, or when to change gears from building power to going all-out for victory points in many Euro games.
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2012/04/19

Fun with probability – Part III

Filed under: Gaming Blog — Tags: , , , , — AlexWeldon @ 9:28 pm

Over last couple of days, I’ve been working on analyzing a simplistic game that I came up with to talk about risk-reward decisions in multiplayer games. What I thought would lend itself to easy analysis in order to prove a point, however, turned out to be a pretty complex and interesting math problem. Yesterday, I presented my findings for the two-player case. As you’d expect, it gets a lot more complicated when you add a third player; so much so that I didn’t even bother trying to work anything out for a four-player situation.

The first interesting thing to notice is an elaboration on what I said previously, about larger die sizes (and thus a larger range of choices) favoring the player who gets to pick last. When we think about multiplayer games, we can see that the actual concern has to do with the number of choices relative to the number of players; the extreme case would be that in which we have as many players as there are sides on the die. In that case, we know that all numbers will be chosen in the end. Thus, the first player has just as much information as the last, and can therefore choose the best number for himself, meaning that the last player is at the greatest disadvantage.

In the three-player case, it (perhaps surprisingly) turns out that the break-even point is once again that of the standard six-sided die. The first player should choose 4, the second should choose 5 (just as in the two-player game) and the third is now left with no better choice than to pick 1 and hope the other two fail. Thus, the second player has a 1/3 chance of winning outright, the first player will win 1/2 of the 2/3 of the remaining times, thus 1/3 as well… leaving 1/3 for the third player.

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Fun with probability – Part II

Filed under: Gaming Blog — Tags: , , , — AlexWeldon @ 2:45 am

Yesterday, I posted about a little thought experiment game I’d come up with to look into risk-reward decisions in multiplayer games.

In the game, each player in turn picks a number, from 1 up to the highest number on whatever die is being used. Then everyone rolls, trying to get their number or higher. Out of those who succeeded, the one who picked the highest number (i.e. who took the biggest risk) wins. If everyone fails, they all reroll until at least one person succeeds.

It’s easy enough to work out some basic results for the two-player version on paper. Yesterday, I posed six questions of increasing difficulty to be answered, whether mathematically or simple guesswork. Here they are again, now with the answers.
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2012/04/17

Fun with probability – part I

Filed under: Gaming Blog — Tags: , , , — AlexWeldon @ 7:40 pm

A friend of mine just posted on my Facebook wall, linking to this YouTube video about “Grime Dice,” a set of five dice with numbered faces chosen to have some interesting non-transitive properties; the first is that each of the dice will statistically beat two of the other dice, forming two “A beats B beats C beats D beats E beats A” loops, like Rock-Paper-Scissors-Lizard-Spock. The second, more remarkable property, is that if you roll two dice at a time instead of one, and add the totals, one of these loops remains unchanged, while the other reverses in order (so that E beats D beats C beats B beats A beats E).

After writing my last post, about how risk-reward decisions are affected by a game in which the goal is achieving an all-time high score, I got to thinking about more general cases of risk-reward decision-making in games, and how that is, like these Grime dice, a non-transitive thing. If you have the opportunity to see what kinds of risks your opponents are taking, you’re usually going to want to gamble either just a little bit bigger, so as to come out slightly ahead if you both succeed, or – if you feel your opponent’s strategy is too high-risk, play as safely as possible and count on them failing.

Having been reminded of this by the Grime dice, I decided to invent an extremely minimalist dice game to take a closer look at this idea in the abstract.
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