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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2012/05/01

Fun with Probability – conclusions

Filed under: Gaming Blog — Tags: , , , — AlexWeldon @ 4:07 pm

I’ve been busy the past couple of weeks, and failed to summarize the general conclusions I think can be drawn from my analysis of the simple dice game I proposed. But I’d like to do so before I forget about it entirely.

As a refresher, the basic premise of the game is that players, in turn, get to pick what number they’re going to try to roll on a die. If only one player gets their number, they win, but if two or more do, the one who picked the harder number to roll is the winner.

The conclusions that I reached from my analysis, and which I think have general application to most games in which players get to choose their level of risk, are as follows:

  1. It’s never correct in this game for anyone but the last player to choose a very safe (better than 50% shot) number. The reason is that someone choosing afterwards can always take just a slightly bigger gamble (but still better than 50%) and win most of the time. What this means in general for games of this sort is that you don’t want to play it too safe until you know what your opponents are doing – and even then, only play it safe if you feel they’re all likely to fail. Games are not for the risk-averse!
  2. In a two-player game, you always want to either push just a little harder than your opponent, or else play it as safe as possible if you think he’ll fail. This kind of brinksmanship is intuitive, but the key strategy in most games of this sort would be in determining exactly where that brink lies. In the simple case of a game where the bigger gambler wins if both succeed and you keep going if both fail, you’re shooting for around a 40% chance of success.
  3. When playing with more than two players, you don’t want to match anyone else’s strategy too closely if others still have a chance to adjust theirs. This is the least intuitive results, as gamers often fall into “groupthink” patterns, wherein everyone plays a similar strategy. But it makes sense when you think about it; if two people are doing the same thing, the third player is effectively playing against a single opponent (albeit one who gets two shots at succeeding), and it’s thus easier for him to pick a winning counter-strategy. If the opponents vary their strategies, it’s hard for the remaining player to find a single counter-strategy that works against both.
  4. When your opponent gets a chance to react to your strategy, your best move is generally the one which puts him in a position where all choices are equally attractive. When there’s little advantage to choosing one strategy over another, you minimize the advantage of having that choice.

These are interesting conclusions, and intuitively correct once they’re pointed out. The third – about adopting different strategies than your opponents rather than imitating – is the most interesting of them, and will probably merit additional investigation another day.

Related: Fun with Probability – Part I

Related: Fun with Probability – Part II

Related: Fun with Probability – Part III

2012/04/19

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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