I just read the rules for a new (to me) game on Yucata: ConHex. The game itself looks to be an interesting 2-player abstract game but what really caught my eye was the rule they use to start the game. They call it the pie rule because it's how you get siblings to split a piece of cake and I guess cake and pie are synonymous.
The first player makes whatever move he wants. (Typically in an abstract game going first is a big advantage. Think tic-tac-toe and the middle square if that game wasn't a guaranteed draw.) Then the second player can either take a normal turn or he can remove the first player's piece and replace it with his own.
There may still be an advantage to going first (or to going second) so there's still some 'luck of the draw' going on but assuming both players know what they're doing that advantage is going to be minimized. The guy going first can't risk taking the 'best' spot since he'll just end up giving it to his opponent. It depends on the specific game but it's entirely possible that he can't risk taking the 'worst' spot either since it may be a negative EV play and his opponent will just let him keep it. So ideally he's going to find a move as close to even as possible...
If the game has a full spectrum of plays from good to bad then it also adds in predicting how likely your opponent is to swap. If you think he'll swap more often than he should then you can pick a worse starting spot expecting him to switch in. If you think he'll swap less often than he should then you can carve out a better starting position for yourself.
Even if it doesn't, and any opening move is good, there's still going to be a worst spot and you can force your opponent to swap into it. This at least helps deal with an unbalanced starting position!
6 comments:
I was discussing Conhex the other day with Robb. And we both felt that even with a very weak first move it is good to be the first player. This means that the second player should always switch.
When dividing pie, you can make a 49-51 split. In deterministic games you can only split it 0-1 or 1-0. The question is how hard is it to decide who is in a winning position?
I've only played 1.5 games so I don't really know much about the game but it does seem like switching is always right. So I guess first player should just take one of the corner spots?
They centre spot is also a weak option. It also only touches one space, though it is1/5 vs 1/3.
I'm not convinced you can play on the other side of the map, so I think corner spots may be strong.
Not knowing anything about the game at all, just thinking about the theoretical idea of the pie-goes-first mechanic, it does seem very difficult in a deterministic game.
Consider any game which is solved (or even just provably solvable). If the game is second player wins then obviously this rule just sucks for the first player. If the game is a tie then the rule changes nothing. If the game is first-player-wins then the rule either changes the game to a tie or makes it second-player-wins if all opening moves can result in a win.
Given this admittedly simple analysis, I'd say that pie-goes-first makes it obviously better to be second player than first, so it's not really dividing the pie (where assuming a reasonable amount of cutting skill both people should be happy).
I think the rule is really neat, and if the game is sufficiently complex then it might be a great rule. But I'm pretty sure it doesn't solve the problem of first player being better, it just makes second player better instead.
I agree with you completely. In a solved game under perfect play the rule doesn't really do anything. Someone won the coin flip and therefore wins the game and we've merely changed which of heads or tails won. (Or maybe made it so no one wins.)
What it does do is make a game more balanced before it gets solved or when the players don't play perfectly. All abstract games are really just determined by starting position but people still play them and I think this rule makes for 'fairer' games before perfect play starts kicking in.
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