Game Theory

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Liar’s Dice Addendum

My math-minded brother has some additional thoughts to add to the Liar’s Dice Strategy. I’ve asked him to guest blog about them.

If you want to get nerdy you can add the following when calculating a bid:

Most players use probability when they play: the number and 1’s = 2 options out of six = 1/3 on average for any number. This would suggest if there are 15 dice in play, there should be around 5 of each number.

What some people don’t take into account accurately is their own dice that they already know. With this knowledge, you can use conditional probabilities to calculate a “riskier” bid while still being relatively safe. Essentially, you use the information you already have to calculate how many potential dice there are.

Say there are 3 players, 5 dice each (15 total) anyone should feel pretty safe calling 4 or 5 of any number. However, you know 5 of the dice already, so you should adjust your math! You only don’t know 10 dice, so use the 1/3 estimate on those dice and add in your own. If you have 4 sixes (combination of 1’s and 6’s), then estimate 2-4 sixes under the other players’ cups and add your own 4. That way you can pretty safely bid 6-8 sixes (I’d probably go 7 myself to be a tiny bit safer), which seems like half of the dice.

When someone does the math quickly, 7/15 ~ 50%, a red flag goes off in their brain. They will likely call Dudo. With your 4, it isn’t too hard to get the 7 you need, since each player will likely have 1-2.

Sometimes, you have to take the defensive. Listen to what other players are bidding. If they’re only bidding fives, and all you have is sixes, this may be a good round to take it to ones. Or, use your knowledge of the 5 dice you know (0 fives), and estimate based on the other players’ bids. There can’t be more than 10 fives if you don’t have any, and there are probably more like 7 if they both seem really confident.

This method isn’t foolproof, due to variability and other people having knowledge of their own dice, but you can get the people around you to lose a lot more dice, or at least make them really uncomfortable raising the number, which is where the real fun comes into play.

Be aware, when you are calling your bids, that there is more variability the fewer dice there are (which is how probabilities work: small N, larger variability), so if your dad says “Oh man! Holy moley!” and starts giggling, you should probably just guess he has at least 4 of whatever he’s laughing about and go from there.

Thanks Nathan, great tips.

1 Comment Posted on Game Theory, Strategy Games

Blokus Strategy 2 – Bayesian Probabilities

This piece of Blokus strategy deserves its own post.

Let me introduce you to the concept of Bayesian probabilities. The math is pretty heavy, so I will just describe the concept.

In regular probability, you look at the frequency of how often something occurs. This works in playing card games. “What is my chance of drawing a diamond”.

Bayesian probabilities assigns probability based on an states of knowledge or belief. “What is the chance of someone playing in that square, given the other options they have to play”

This is pretty complex, so let me give you an example in Blokus. Suppose there are 4 different places that you are thinking about playing on the board. When you look at each one, the you gather the following information about each option.

  • Option 1: No one else can play on it. It is a perfect fit for your square piece
  • Option 2: All 4 players can play into this space. Once played there, it is isolated, so it won’t get you anywhere else.
  • Option 3: Only you and one other person can play here. The other player needs this space to avoid getting stuck.
  • Option 4: You and 2 others can play here. It leads to one other open space

How do you decide what to do? Bayesian probability tries to assign probabilities to each of beautiful-mindthese sites to decide the probability that you won’t be able to play there later. It seems complex, but your brain is already doing this when you think through where you should play. I’m going to assign probabilities the way my brain naturally thinks about it, without actually calculating probabilities. (No one has time for you to get your whiteboard out and Beautiful Mind the actual probabilities.)

  • Option 1: 0% probability of losing this space before your next turn. Wait for a later turn to play here
  • Option 2: 80% probability of losing this. Potential for this turn
  • Option 3: 90% chance of losing the space. Would really hurt the other player, and you won’t have a chance later
  • Option 4: 85% chance of losing the space. This will give you another play, so if you need the extra space, this may be your best choice.

At this point, you know you will probably lose two of the spaces, so do you want to use the chance to block someone out, or to sneak into extra space? Its up to you, just go in understanding the consequences.

Are there any other techniques you use to figure out where to play? Leave a comment.

5 Comments Posted on Game Theory, General Strategies

Strategy for your Strategy

When I was young, my older brother was two years older than me and always beat me in games. Once, he created his own game and didn’t tell me any of the rules. While he rolled dice, chipping away at my base, I sat there in confusion. Eventually, he said, “aren’t you going to defend yourself?” At that point, I lost it. I jumped across the game board and attacked my brother. (I’ll show YOU defend yourself). Unfortunately, he was also bigger and stronger than me, so it didn’t end well. This event helped to shape me.

So you want to win at games? Tired of losing to your older brother, next-door-neighbor, boyfriend, girlfriend, neighbor?

You need a strategy. And to start with, you need a strategy of how you are going to get a strategy. Here are some strategies to develop your strategy.

  1. Always read the rules BEFORE the game
  2. Pay special attention to the winning conditions
  3. Set yourself up early for success later in the game
  4. Learn from your failures
  5. Be positive.

Read the rules

When you show up to a game night, often people will suggest a game you have never played. This isn’t a problem, but if you use the game as your “learning” game, you won’t win. Also, if you count on someone else to explain the game to you, you miss out on all the tiny details of how to win and how cards and pieces and players interact. You don’t have to take half an hour to delay everything, just take some time to read them.

Winning conditions

Winning conditions matter. Some games end as soon as the last card is drawn. Some end when all your guys are dead. Sometimes you get extra points if you held onto the small piece and played it last. You need to know how to win to be able to win.

The long game

It doesn’t matter if you get up a little at the beginning of the game if you don’t plan a strategy that will last through the whole game. Figure out the following:

  • What should you do at the beginning to set your self up for the remainder of the game
  • What is your interim strategy, how do you you sustain yourself through the game to get to the end in a position to win
  • Towards the end, when do you need to make the moves to end the game or get the most points possible before it ends.

Learn from your failures

You won’t win every time. Maybe you planned out a strategy that you thought was great. Maybe it really was great, but someone else got lucky. Or maybe you need to tweak your strategy. That is OK. Don’t jump across the board and attack. Just say, “Good game”. BUT, that isn’t the end. Keep tweaking your strategy until you figure out what works well for you to win

Win and lose well

If you win, tell everyone they played well. If you lose, tell everyone they played well. There is a delicate balance between dominating the competition and being a fun person to play with. If you aren’t fun to play with; eventually, no one will play with you.

And, it is the right thing to do.