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Subject: Kingmaker rss

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Dr. Gordon Hamilton
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Is it possible to have a PURE-STRATEGY game with 3 players?

When I call a game a PURE-STRATEGY, I mean that infinitely intelligent players know the outcome before they start the game. Chess, Go, and Dvonn are PURE-STRATEGY games because infinitely intelligent players would get bored playing because every time the outcome will be the same – win for white, win for black or a draw.

So is it possible to have a PURE-STRATEGY game with 3 players?

Yes. Example: Tic-Tac-Toe where the objective is to get 2 pieces in a row is always won by a reasonable first player. But that is not a FUN game for adults. I define a game as FUN if its outcome is not guaranteed for the players. So tic-tac-toe is FUN for 3-year-olds, but not adults, and chess is FUN for adults, but not infinitely intelligent players)

Is it possible to have a PURE-STRATEGY game with 3 adult players which is FUN?

Yes, but such a game cannot be played! Proof: In order to make the game PURE-STRATEGY a rule must get rid of the king-maker problem. The king-maker problem occurs when one player is forced to make a choice about which other player will win the game. However, this rule that solves the kingmaker problem requires the players recognize when the king-maker problem exits, and this requires infinitely intelligent players. Ordinary mortals cannot even play the game because it will be unclear to them exactly when such a rule should be applied.


Okay, so I know that it is impossible to create a playable purely strategic version of a game for 3 players, but help me find a compromize. One idea is to introduce a rule to stop the kingmaker problem:

All players must play for a win, but if they cannot win, a player must play to keep the game going for as long as possible.

For example if Allen, Betty and Carl are playing in that order…

• It is Allen’s turn. Betty and Carl both have winning moves on their next turn. If Allen can stop Betty… then Allen must stop Betty.
• It is Allen’s turn. Betty does not have a winning move, but Carl has a winning move on his next turn... then it is illegal for Allen to give Betty a winning move unless he has no other option. It is also illegal for Allen to let Carl win if he is the only one who can stop him.


As shown above, it is impossible for players to know every time this rule applies, so a three-player PURE-STRATEGY game with such a rule turns into a conversational game where players try to convince others when the rule applies and the righteousness of their opinion.

I love playing with this type of rule. It ironically turns a PURE-STRATEGY game into a light-hearted free-for-all that academic thinkers will enjoy, but cannot dominate.

I want to hear what people think of this compromize. The results will be of practical import as I plan to use them in a future release of Santorini. If people do not have one of the 60 existing copies of Santorini, the same discussion applies to all PURE-STRATEGY games seeking to become multi-player.

Sincerely,
Gord!
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Stephen Tavener
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Gord!,

Given what you've just said, I think you should reconsider your low rating of Cosmic Encounter... that's a large part of what the game is

I think Santorini is always going to have problems 3-player; take a look at the following example... vastly simplified, but this kind of thing can happen in real play:

0 0 2A 3 A
0 0 0 0 0
0 0 C C 0
0 0 0 0 0
0 B 3 2B 0


Players are A, B, C; A and B can win next move, C (to move) has a simple choice; cap A's tower, allowing B to win, or (anything else), allowing A to win. This is never a pleasant position for C to be in.

In general, there are always problems of this type in a 3-player game; from a more theoretical viewpoint, if all 3 players go into the game with equal expectations of winning (1/3), then it is clear that with two players collaborating, the other player has a low expectation of winning (2/3 vs 1/3). That means that you can't play to win using strategy alone.

The closest a pure strategy game has come to working 3-player is this one:

http://www.boardgamegeek.com/game/11632

... in effect, the scoring mechanism means that the player who does the best job of beating up on the player to their left will win the game. You still aren't really entitled to feel that you won through superior skill, but you _are_ entitled to feel that you lost through inferior play!

Another way to remove the kingmaker problem would be to make the game a partnership game to some degree... one player(A) gets two pieces, one power; the other players B, C get one piece and a power each... turn order is A B A C.
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Michael Van Biesbrouck
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If you believe that all of your opponents are going to gang up on you without concern for who wins (as long as it isn't you), then you are only going to play a multiplayer game because you want a big challenge.(1) Unfortunately, even if your opponents are self-interested, they aren't necessarily optimal opponents and may make mistakes that favour one player more than another.(2) Even given excellent play, an opponent may have to choose which of two opponents to harm more on a particular move. The unpredictability of opponents also makes it impossible to choose optimal moves in general.

What we can try to do is move unsatisfying decisions further away from the end-game. If one player significantly harms you in the mid-game, then you can force the other two players to police each others' footing until you have repaired your own position. (If they are not in a situation where balance can be maintained, then a winning move was made rather than an `unfair' one.) (3)

In my game package, Gord! sent me two possible anti-king-making rules. One of them is described above. The other one was quite complicated. I found them both unsatisfatory and suggested two alternatives. They are both intended to address the final moves of the game, although indirectly they will affect strategy.

Tie rule:

If both other players can win before your next turn and your move will determine which one wins, then those two players have tied. Players should always attempt to
win; failing that, to tie for a win.

If you like to keep score, tied players get 0.5 points and winners 1.0 points.

Paranoid rule: (4)

When possible, you cannot leave the board in a state guaranteeing a win for either of the other players. If you leave the board in a state that allows the next player to king-make between you and the third player, then you come in second, the king-maker is third and the remaining player wins.

I think that this rule is almost the antithesis of Gord!'s `option B', which he might want to present here.

Footnotes:

(1) There is a great story on BGG about a family that regularly played Civ. The father was so good that everyone else had to gang up on him to prevent him from always winning.

(2) Puerto Rico with inexperienced players, anyone?

(3) Responsibility for blocking wins in a multiplayer game leads to discussions such as those on the Attika forums.

(4) Yes, everyone is out to get you.
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Dr. Gordon Hamilton
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mrraow wrote:
I think Santorini is always going to have problems 3-player;


As shown above, all FUN 3-player PURE-STRATEGY games are formally unplayable because it is never clear when the rule to stop the kingmaker problem should be inforced. Gute Nachbarn, the game you pointed me to, is not a PURE-STRATEGY game, because the kingmaker problem still exists.

mrraow wrote:

0 0 2A 3 A
0 0 0 0 0
0 0 C C 0
0 0 0 0 0
0 B 3 2B 0


According to the rule presented above, C cannot win, so is forced to make the game as long as possible. This means C must stop A. B wins this game.

mrraow wrote:
Another way to remove the kingmaker problem would be to make the game a partnership game to some degree... one player(A) gets two pieces, one power; the other players B, C get one piece and a power each... turn order is A B A C.


That's a two-person game in disguise ;-)
 
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Dr. Gordon Hamilton
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mlvanbie wrote:
I think that this rule is almost the antithesis of Gord!'s `option B', which he might want to present here.


Here it is... rule B:

To lessen the kingmaker problem in 3-player Santorini all players must play for a win (with one exception), but if they cannot win, they must play to keep the game going for as long as possible.

For example if Allen, Betty and Carl are playing in that order…

• It is Allen’s turn. Betty and Carl both have winning moves on their next turn. If Allen can stop Betty… then Allen must stop Betty.
• It is Allen’s turn. Betty does not have a winning move, but Carl has a winning move on his next turn... then it is illegal for Allen to give Betty a winning move unless he has no other option. It is also illegal for Allen to let Carl win if he is the only one who can stop him.
The one exception: It is Allen’s turn. Betty does not have a winning move, but Carl has a winning move on his next turn which Betty can stop... then it is illegal for Allen to create a winning move for himself that Betty could stop (if she did not have to stop Carl’s winning move).

Either rule A or B solves the kingmaker problem and makes Santorini PURE-STRATEGY, but makes the game formally unplayable because it is impossible for mortals to know when the rule applies.

The tie rule and Paranoid rule do not solve the kingmaker problem, but just pushes it back 1 move. Such games would not be PURE STRATEGY.
 
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Stephen Tavener
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Ah, Clementine is in bed now, so I can respond to this thread with only the cats clamoring for attention; sharp claws, but ultimately, less distracting. A few years ago, a friend introduced me to his kingmaker rule:

A player is a kingmaker if:
- it is their turn
- they cannot win
- the game is about to end
- the player may choose which of at least two other players wins according to the normal rules of the game.

Anybody finding themselves in this situation may claim a win; however, they must be able to prove to the satisfaction of all other players that these conditions apply.
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Stephen Tavener
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Quote:
According to the rule presented above, C cannot win, so is forced to make the game as long as possible. This means C must stop A. B wins this game.

Sorry... I should have read your post more carefully before replying. I still think that under these rules, C is not going to enjoy the game; not only is C stuck in a game that C canot win, but C is forced to prolong the game for as long as possible.
 
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Michael Van Biesbrouck
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Games by Gord! wrote:

The tie rule and Paranoid rule do not solve the kingmaker problem, but just pushes it back 1 move. Such games would not be PURE STRATEGY.


I believe that my two rules push the kingmaker problem much further back than a single move, as they affect which board positions that players will be willing to enter. Anyone planning ahead several moves will not enter the suggested board position unless all alternatives are worse. In the endgame it should be possible to fully evaluate the consequences of many possible moves. I think that the consequences of these rules go back as far as the players can fully evaluate board positions.

In the example, player B would only enter the given position if a tie/second was the best situation. If a win were possible, then B would make a non-check move, forcing C to block A. Similarly, A would not move into the shown position under the tie rule if there were better options than a tie. (Obviously, if A's move leads to B winning through a move that forces C to block A, then A would never move into that position anyway.)

Both of my rules prevent a player with no hope of a win/tie/second place from flipping a coin/soliciting bribes at the last instant. At an earlier position, the player might be presented with such a choice. On the other hand, neither A nor B wants to put C in such a position. The midgame should consist of maneuvering to avoid such uncerntain outcomes until one player, through better lookahead, realizes that a particular move guanantees a best-possible finish and siezes the opportunity.

For any three-player game without randomness or simultaneous action selection, we should be able to label each position as a win for a particular player, kingmaker position for a particular player or stalemate. Santorini doesn't have inifinite move sequences without gods, but three-way losses are possible. (Adding ties for first and second through my rules does increase the number of labels.) So we may find that A always wins or C kingmakes. Possibly B decides whether A or C kingmakes. Given a complete understanding of the game, the game will hold no interest. The actual point at which one player kingmakes may be so early in the game that we happily pass it and make non-optimal moves that undo the kingmaking. Exploring these possibilities non-exhaustively is the essence of games without non-determinism. (It is provable that once a deterministic game exceeds the players' ability for complete evaluation then it is possible to add non-determinism -- the outcome of move sequences can be as random as a coinflip due to lack of computational power.)
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Michael Van Biesbrouck
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mrraow wrote:

A player is a kingmaker if:
- it is their turn
- they cannot win
- the game is about to end
- the player may choose which of at least two other players wins according to the normal rules of the game.

Anybody finding themselves in this situation may claim a win; however, they must be able to prove to the satisfaction of all other players that these conditions apply.


I also considered this rule. Unfortunately, the `normal rules of the game' part is a serious weakness as it eliminates the kingmaker rule from consideration. Recursively applying the rule is also problematic. We end up with players that can make either of the other players kingmakers yet cannot win themselves.

I once played a game of Ad Acta in which the last-place player made a surpise move as the last move of the game that totally rearranged the scores of all of the other players. The game's owner, robbed of his anticipated win, was somewhat unhappy about this. I made a comment along the lines of the above rule, but I am left convinced that this rules is also unsatisfactory to many players. Nonetheless, I feel that rules should encourage players to avoid making someone else a kingmaker, hence my suggestions.
 
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Michael Van Biesbrouck
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I played another three-player game today (and two two-player ones). Kingmaker issues came up quickly (and were commented on by a new player), but we found it an enjoyable challenge to work around those issues and force the other players to block one-another while we worked towards a win. This was satisfactorily mind-expanding; once the players knew what they had to work against (and with) we were ready to have a fun game.
 
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Lyman Hurd
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What has been discussed is a specific type of endgame. However, in the course of maximizing my position over the course of an entire game chances are that my moves will have positive and negative effects on my two opponents. Assuming that I rationally make the move each turn that most increases my utility function, the landscape faced by the other players is inherently not "fair". I do not need to have an intent of favoring opponent A or B to actually have the effect of favoring A or B.

The other extreme is to take the tack of Gute Hachburn and give people a rationale for favoring one opponent over another. In that game you get points for the following player's accomplishments. In the current situation I think it woul dmake more sense to tie someone's fortunes to the preceding player as it is harder to help someone two moves away hence prolonging the game.
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Clark D. Rodeffer
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mrraow wrote:
A few years ago, a friend introduced me to his kingmaker rule:

A player is a kingmaker if:
- it is their turn
- they cannot win
- the game is about to end
- the player may choose which of at least two other players wins according to the normal rules of the game.

Anybody finding themselves in this situation may claim a win; however, they must be able to prove to the satisfaction of all other players that these conditions apply.

I got to thinking about kingmaking last night, in particular with respect to three-player Santorini. Not previously having read this thread, I came up with what mostly amounts to what Stephen relayed here.

In a multi-player pure strategy game, such as three-player Santorini, if one player can clearly demonstrate that any two or more other players have somehow conspired, whether consciously or not, to place him in a king-making situation, they must, whether consciously or not, really want him to be king. After all, they're giving him the power to decide, which is what kings do. Therefore the "kingmaker" deserves the win.

For example, imagine a three-player pure strategy game (such as Santorini, but it could be almost anything else) between players A, B and C, who take turns in that order. Suppose it's player A's turn. Player A immediately wins if he can clearly show all of the following:

1) On his turn, player A cannot make any move that can immediately satisfy any of the normal game winning conditions. In Santorini, this would be climbing to the third level, eliminating or preventing all opponents from moving and/or any other special winning conditions provided for by his godly power. This first part isn't strictly necessary, because if player A could immediately win, he should do so and be done with things. But it does set up a prerequisite condition for 2 and 3 below.

2) On his turn, one or more moves that player A could legally make would produce situations whereby player B could win on his very next turn, and ...

3) ... all moves that player A could legally make to prevent player B from winning on his very next turn, at least one of which must be shown to exist, also have the effect of preventing player B from making any moves that could prevent player C from winning on his very next turn.

Granted, this does introduce a third winning condition to the three-player game, that of proving yourself to be in a kingmaker situation. But is that really so bad? Winning under such a condition may not feel as satisfying as winning by climbing to the third level or whatever alternative winning condition your godly power might allow. But shouldn't the threat of that happening be enough such that all three players will work to prevent it happening?

Clark
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Lyman Hurd
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I was playing a three player game of Santorini recently and it occurred to me that the "extend the game as long as possible" heuristic had a flaw at least if considered only one move deep.

I was Player A and I reached a position in which Player C could win on her next move and I had to decide between:

1) Preventing Player C's win.

2) Setting up a win for myself leaving Player B in a position to foil either me or Player C but not being able to do both.

If Player B takes the attitude that the game needs to be prolonged as long as possible, she should allow me to win. However, given that I could have stopped Player C myself and failed to do it, it would certainly feel to me as if I should lose.

Of course this was all moot speculation as I really took course of action 1).
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Michael Van Biesbrouck
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I think that part of the issue is play style. If someone observes that a kingmaker situation has been reached, then we check the board for alternate moves and say that the game is over. If we see that a player has moved to give another person an immediate win then we often roll back the move and find a more challenging one. (In a recent game we didn't roll back the move where I forgot the next player was Hermes, though. )
 
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Dr. Gordon Hamilton
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lyman wrote:
I was Player A and I reached a position in which Player C could win on her next move and I had to decide between:

1) Preventing Player C's win.

2) Setting up a win for myself leaving Player B in a position to foil either me or Player C but not being able to do both.

If Player B takes the attitude that the game needs to be prolonged as long as possible, she should allow me to win. However, given that I could have stopped Player C myself and failed to do it, it would certainly feel to me as if I should lose.


Two points Lyman:

First,
It was actually player C, who, on the previous turn, made a mistake and handed the game to you.

Second,
This was the exact situation which option B (see above) was designed to stop.
 
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Paul Boos
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I'll make some comments - Lyman's last comment presumes one thing, which (at least in Santorini) is generally not the case IMHO. This means Player B (I think if I followed it right) could not prevent... Thisthe preventing a win can pass around. If you miss it, then it was just your own bad analysis. Of course, I always expect collaboration to occur. But you forgoing taking action to defer it to the next player is natural play.

I see the problem as one of normal game play. Obviously trying to minimize the ease to which the situation can arise is a good idea, but there is no problem with it happeneing. I just feel if you make special exceptions too often, then the game wil become unwieldy.

Given that - if one is looking for such a solution - using the person that has the kingmaking situation becomeing the inner seems to be the best answer. It wil be hard to perfectly prove though, uness the game is simultaneous in its movment mechanism. Because there is always the next player who can try and answer the call.

For Santorini, I see this happening about 50% of the time in a 3 player game. To me that means it is possible to brilliantly play your men so that you can win anyway and yet the game can be close until the end. If this winds up feeling too often for you Gordon, then leave it as a two player game. But I really like the feel in a 3 player game and don't feel this is problematic at this level. It woudl be if the gamne devolved into this at 75% or more of the time...

Paul
 
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Paul Boos
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I really need to proof my typing more often... That was a horrible post.
 
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