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Subject: A Venn diagram to illustrate Abstract Games article on the BGG wiki rss

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Channing Jones
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A game is usually considered "abstract" if it has all three of those properties, even though the name implies only unthematic.

For the luckless but not perfect information and unthematic you can put "Kriegsspiel" by Henry Michael Temple (1899) where two players play hidden chess against each other using an umpire.

For the last category luckless, not perfect information and thematic you can put the classic "Battleship" game.
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Benedikt Rosenau
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Jeffrey Henning wrote:

I found multiple sources that said perfect information is orthogonal to randomness, alleviating that concern of mine.
I wonder who or what these sources are. I can quote standard texts on randomness being an effect of incomplete information.
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J. Alan Henning
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Zickzack wrote:
I wonder who or what these sources are. I can quote standard texts on randomness being an effect of incomplete information.
The texts were more about classifying games. I didn't bookmark them -- sorry. The frequent example was Backgammon.
 
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Matteo Perlini
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Jeffrey Henning wrote:
Zickzack wrote:
I wonder who or what these sources are. I can quote standard texts on randomness being an effect of incomplete information.
The texts were more about classifying games. I didn't bookmark them -- sorry. The frequent example was Backgammon.
Maybe you are referring to this article, where the author write:
Quote:
Surprisingly perfect information games may contain random elements. Surely the outcome of the dice roll is information that is hidden from me? Game theory circumvents this problem by declaring the random element – be it a dice roll, a card draw or something else – as something that occurs outside of your turn and is thus part of the game state. Games without any kind of random element are known as deterministic.

So, enough of the theory, everything is easier with some examples: Chess, Go, Quarto and tic-tac-toe are all deterministic perfect information games. There is no randomness involved and the whole game state is visible. As an example for a non-deterministic perfect information game, take Backgammon (or take Senet from last week): there is an element of chance, but all information about the game is visible to you.

Of course this is false. An informal definition of perfect information is this one:
"A sequential game is one of perfect information if only one player moves at a time and if each player knows every action of the players that moved before him at every point. Intuitively, if it is my turn to move, I always know what every other player has done up to now."

For modelling a non-deterministic game you introduce a fictitious player, called Nature, who selects her strategies randomly, based on some predetermined probability distribution. So the real players don't know all the history of the game (where they are in the game-tree), there are some hidden information, so the game is one of incomplete information, therefore not a perfect information.

Perfect information games have no random elements.
 
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Russ Williams
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epicurus wrote:
An informal definition of perfect information is this one:
"A sequential game is one of perfect information if only one player moves at a time and if each player knows every action of the players that moved before him at every point. Intuitively, if it is my turn to move, I always know what every other player has done up to now."

For modelling a non-deterministic game you introduce a fictitious player, called Nature, who selects her strategies randomly, based on some predetermined probability distribution. So the real players don't know all the history of the game (where they are in the game-tree), there are some hidden information, so the game is one of incomplete information, therefore not a perfect information.

Perfect information games have no random elements.
I am confused by your explanation. If (e.g.) I randomly determine what my moves will be in a chess game with you, are you saying that this game of chess is now no longer a "perfect information" game?

Just because Nature decided randomly doesn't mean we don't know what Nature's decisions were. Similarly I may not know why you decided to move your knight to g3 but that doesn't mean I don't know that you moved your knight to g3.

Or to put it another way, the "hidden information" you mention in your Nature example seems only to be the thoughts/motives of the players/Nature, not the publicly visible game state and actions themselves, which is surely all that should matter, otherwise every game has "hidden information" since we cannot read our opponents' minds.

What am I missing?
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Matteo Perlini
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russ wrote:
I am confused by your explanation. If (e.g.) I randomly determine what my moves will be in a chess game with you, are you saying that this game of chess is now no longer a "perfect information" game?
If you are playing a chess-like solitaire, a puzzle, where the enemy pieces move randomly, this one is not a "game" in game theory terminology.

A game is "the interaction among rational, mutually aware players, where the decisions of some players impacts the payoffs of others. A game is described by its players, each player's strategies, and the resulting payoffs from each outcome."

For studying your puzzle game you should use the "decision theory".

russ wrote:
Or to put it another way, the "hidden information" you mention in your Nature example seems only to be the thoughts/motives of the players/Nature, not the publicly visible game state and actions themselves, which is surely all that should matter, otherwise every game has "hidden information" since we cannot read our opponents' minds.
Consider a chess-like 2-player game where you first choose which piece to move and then you roll a die to determine how many steps you can do with that piece. This is hidden information. All the die rolls can be predetermined before the game starts, written down on paper and revealed few at the time. It is the same.
 
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Benedikt Rosenau
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Jeffrey Henning wrote:
Zickzack wrote:
I wonder who or what these sources are. I can quote standard texts on randomness being an effect of incomplete information.
The texts were more about classifying games. I didn't bookmark them -- sorry. The frequent example was Backgammon.
Question: a pack of cards has been shuffled. What is the chance that the topmost is an ace of spades? Answer: barring Schrödinger's cat, it either is an ace of spades or is not. It is not a question of probabilities. However, if the question were what a fair bet would be, 1/52 is the answer.

I hope this example explains how incomplete information leads to calculating probabilities.

So, in a game there can be things both players do not know (like the results of rolls) or things only one player knows - or both forms of incomplete information. You can classify games by that.
 
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Russ Williams
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epicurus wrote:
russ wrote:
I am confused by your explanation. If (e.g.) I randomly determine what my moves will be in a chess game with you, are you saying that this game of chess is now no longer a "perfect information" game?
If you are playing a chess-like solitaire, a puzzle, where the enemy pieces move randomly, this one is not a "game" in game theory terminology.
Fine, consider a 3-player competitive game with no random mechanisms or hidden information in the game rules (let's say Blokus Trigon, for example, or RED, or Yavalath) where one of the players is moving randomly.
You seem to be saying that this is not a game, and that the players don't have perfect information about the game history.

Or consider playing correspondence or online chess, where you simply don't know how I'm making my decisions. Maybe I'm thinking and trying to make rational moves; maybe I'm making random moves. Does this put the game into some kind of "Schrodinger" or "Heisenberg" state of being "perfect information" and "not perfect information" at the same time?

I don't see how the fact that one of the players is (or might be) playing randomly means I don't have perfect information about what moves they have made or that the game is not a "perfect information game".
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Martin G
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Zickzack wrote:
Question: a pack of cards has been shuffled. What is the chance that the topmost is an ace of spades? Answer: barring Schrödinger's cat, it either is an ace of spades or is not. It is not a question of probabilities. However, if the question were what a fair bet would be, 1/52 is the answer.
Go read up on Bayesian statistics!
 
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Maurizio De Leo
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russ wrote:
epicurus wrote:
An informal definition of perfect information is this one:
"A sequential game is one of perfect information if only one player moves at a time and if each player knows every action of the players that moved before him at every point. Intuitively, if it is my turn to move, I always know what every other player has done up to now."

For modelling a non-deterministic game you introduce a fictitious player, called Nature, who selects her strategies randomly, based on some predetermined probability distribution.

If (e.g.) I randomly determine what my moves will be in a chess game with you, are you saying that this game of chess is now no longer a "perfect information" game?
Just because Nature decided randomly doesn't mean we don't know what Nature's decisions were. Similarly I may not know why you decided to move your knight to g3 but that doesn't mean I don't know that you moved your knight to g3.

Or to put it another way, the "hidden information" you mention in your Nature example seems only to be the thoughts/motives of the players/Nature, not the publicly visible game state and actions themselves, which is surely all that should matter, otherwise every game has "hidden information" since we cannot read our opponents' minds.

That's a very interesting way of describing randomness as separated by information by the introduction of the "Nature" player. This in turn separates "perfect information" from "deterministic" games.
Of course, as every definition, it is tautologically correct, but it is useful only as long as it is shared. The author himself says "And that’s the very short – and probably incorrect, if you ask a mathematician, explanation of perfect information games."

I think the distinction can be useful (and this is why in my Venn diagram the two items are separated in blue and yellow sets).
However an important point is that usually what is meant by abstract games is "Two-player zero-sum perfect information strategy games".
I actually thought of adding the other two dimension to the diagram, but five sets Venn get quite messy.
Maybe I will talk in another post about zero-sum (i.e. non even slightly cooperative). I will briefly mention that Nature "can't win" the game, but let's focus on two-players.
As soon as you introduce more than two players, aspects outside the game take a role: king-making, leader bashing, alliances, and "game politics" in general. By restricting to two players you "abstract" the game from outside considerations.
So being lucky at a two player non-deterministic game can be compared to the player "Nature" being our ally in the game, or "king-making" us.

By introducing the two-player attribute, we can drop the distinction between perfect information and deterministic. I found this kind of taxonomy more useful, but , as said before, it is just a matter of preference in definitions.
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Matteo Perlini
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russ wrote:
Fine, consider a 3-player competitive game with no random mechanisms or hidden information in the game rules (let's say Blokus Trigon, for example, or RED, or Yavalath) where one of the players is moving randomly.
You seem to be saying that this is not a game, and that the players don't have perfect information about the game history.
For the game theory, this one is a two-player game with the Nature. Let the playing order be P1, N, P2. So, N don't play strategically so we can say N has already "chosen" his moves in advance. Therefore, P1 is playing is turn without knowing the payoff of his move, because there are some hidden informations.

russ wrote:
Or consider playing correspondence or online chess, where you simply don't know how I'm making my decisions. Maybe I'm thinking and trying to make rational moves; maybe I'm making random moves. Does this put the game into some kind of "Schrodinger" or "Heisenberg" state of being "perfect information" and "not perfect information" at the same time?
No.
I don't know if I'm playing the Nature or a rational agent, but this doesn't change the objective fact.

russ wrote:
I don't see how the fact that one of the players is (or might be) playing randomly means I don't have perfect information about what moves they have made or that the game is not a "perfect information game".
Because if one player is playing randomly, that particular game (that particular instance of the game) has random elements and for the game theory terminology this is not a perfect information game.
 
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epicurus wrote:
Because if one player is playing randomly, that particular game (that particular instance of the game) has random elements and for the game theory terminology this is not a perfect information game.
I've never heard this definition before. To me, a player is a player, i.e. making moves according to the rules of the game, regardless of how they're making their decision: brute force lookahead in the game tree; heuristic pattern matching; impulsive gut intuition; flipping a coin; asking their grandmother what she'd do; looking up values on an old-fashioned printed table of pseudorandom numbers; intentionally choosing the worst move they can imagine; making whatever move the highest bidder will pay them to make; whatever.

Indeed in classical economic game theory it's often the case that a player's optimal strategy is to choose randomly from among some options, and that certainly does not mean the players are not players.

===

What if you're playing a game of chess and you see 2 moves which seem equally good to you and you see no way to prefer one over the other, so you flip a coin to pick between them. Are you no longer a "player" but just "Nature"?
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Maurizio De Leo
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Quote:
To me, a player is a player, i.e. making moves according to the rules of the game, regardless of how they're making their decision:
I agree with Russ here.
You can't make distinction based on "how" players take decision in games.

It is the same slippery reasoning for which people say chess (among people) has luck. Some say that they choose a move for the wrong reason(for example a miscalculated tactical trick) and then "got lucky" that the move was good for another reason (for example positional consideration).

But nobody really knows "how" exactly they pick moves. We can't introspect enough to understand the process; it comes down to what some people call "intuition", some "luck", some "pattern matching". Computers don't even "know" what chess is or how to play. They just sum numbers based on board states and try to maximize an heuristic function. Some could argue that "Monte Carlo" programs actually play "randomly". But nobody would disagree that they play well and that games against them are "perfect information"

In the end the process of move selection in game must be separated from the definition of the game and from the assessment of quality of play.
The game is still "perfect information" if one player plays randomly, and that player may even be a strong player, based on the results.
Alas, in some cases the "best" strategy might be to pick (pseudo) randomly, like in Ro-Sham-Bo when you are behind.

***********************************************************************

However I disagree with Russ on two examples:
Quote:
intentionally choosing the worst move they can imagine; making whatever move the highest bidder will pay them to make;
If my opponent intentionally tries to lose, then the game is not "zero-sum" anymore. The best outcome for both is my win.
The same applies to being paid from bidders: it may be beneficial for him to do something that "in game" is beneficial for me, dropping the "zero-sum".
This is influence of motives "outside the game", which disqualifies the game from my definition of "abstract strategy" and probably also from the "perfect information". It's ok if I play random moves because I think (even mistakenly) that it's my best shot at winning (in game). It's not if I do it to amuse the spectators (outside game).

In short. Move selection is up to the player, but their goal must be to win the game.

This is also why it's difficult to consider multi-players games purely abstract. When one player realizes that he is out of contention for the victory, he will play based on factors outside the game (like favoring his girlfriend, or attacking the guy who beat him last game).

And this bring us back to the definition of "Nature" as a third player in Backgammon and similar games. "Nature" can not win, and thus has no incentives in-game. She will favor some player based on other factors (maybe fate): the games are in my view not purely abstract.

Sorry for the very long and probably boring post :-D
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christian freeling
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megamau wrote:
If my opponent intentionally tries to lose, then the game is not "zero-sum" anymore.
Nor is it zero-sum if neither tries to win. The distinction between forced cycles and cooperative ones hinges on this. The precence of cooperative cycles doesn't make a game any more 'drawish' because to effectuate such a cycle you must ignore the game's inherent goal.
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megamau wrote:
However I disagree with Russ on two examples:
Quote:
intentionally choosing the worst move they can imagine; making whatever move the highest bidder will pay them to make;
If my opponent intentionally tries to lose, then the game is not "zero-sum" anymore. The best outcome for both is my win.
The same applies to being paid from bidders: it may be beneficial for him to do something that "in game" is beneficial for me, dropping the "zero-sum".
This is influence of motives "outside the game", which disqualifies the game from my definition of "abstract strategy" and probably also from the "perfect information". It's ok if I play random moves because I think (even mistakenly) that it's my best shot at winning (in game). It's not if I do it to amuse the spectators (outside game).

In short. Move selection is up to the player, but their goal must be to win the game.
and
christianF wrote:
megamau wrote:
If my opponent intentionally tries to lose, then the game is not "zero-sum" anymore.
Nor is it zero-sum if neither tries to win.

I would disagree. I think you're mixing the game itself with the "meta" game.

It's quite well defined who wins the chess game, even if one player is trying to get himself checkmated for whatever meta reasons. Perhaps that one player "wins" some meta-game in real life because someone pays him $1000 to lose the concrete chess game, but in the chess game, he has clearly lost. (Indeed, if he hadn't lost the chess game, then he wouldn't have gotten paid $1000 for losing the chess game.)

The forums are full of stories of kingmaking, and boyfriends helping girlfriends, and one guy just playing to hurt another guy whom he is angry at in real life, and so on, in multiplayer games. Many players don't play to win, for whatever reason. I certainly agree that it's typically annoying and unsatisfying to play with someone who's not trying to win, or even intentionally trying to lose or help some other player win or cause some other player to lose. But to me, that doesn't change the fact that we're still playing the game (i.e. following the rules of the game and recognizing when termination is reached and recognizing who won the game - regardless of whether everyone was trying to be that person who won); some of the players are just playing it badly or randomly or to achieve some meta goal like making their girlfriend happy or get paid $1000 for losing or whatever.

I don't think anything about the definition of a game forces someone to try to win. After all in game theory it's well-defined what happens for all possible moves, including obviously bad/losing moves. It's more of a meta social contract that most of us of course find the game more meaningful and fun if we are trying to win, and we would probably choose not to bother playing a game if we knew our opponent was just going to make intentionally bad moves.
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christian freeling
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russ wrote:
christianF wrote:
megamau wrote:
If my opponent intentionally tries to lose, then the game is not "zero-sum" anymore.
Nor is it zero-sum if neither tries to win.

I would disagree. I think you're mixing the game itself with the "meta" game.

...

I don't think anything about the definition of a game forces someone to try to win. After all in game theory it's well-defined what happens for all possible moves, including obviously bad/losing moves. It's more of a meta social contract that most of us of course find the game more meaningful and fun if we are trying to win, and we would probably choose not to bother playing a game if we knew our opponent was just going to make intentionally bad moves.
You're right in terms of formal game theory. I was stepping outside for a moment in favor of a pet subject, namely the effects (or lack thereof) of the mere existence of cooperative cycles in a new game.
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megamau wrote:
However I disagree with Russ on two examples:
Quote:
intentionally choosing the worst move they can imagine; making whatever move the highest bidder will pay them to make;
If my opponent intentionally tries to lose, then the game is not "zero-sum" anymore. The best outcome for both is my win.

Hmm. The masochist who liked a cold shower in the morning so took a hot one.

Despite the valiant attempts at making this complicated, it's actually disappointingly straightforward. The preferred outcome for both players may be my win, but the best (as defined by the terms of the game) is still in each player's case a win for himself.
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Quote:
I think you're mixing the game itself with the "meta" game.
It's quite well defined who wins the chess game, even if one player is trying to get himself checkmated for whatever meta reasons.
Perhaps that player "wins" some meta-game in real life because someone pays him $1000 to lose the concrete chess game, but in the chess game, he has clearly lost.

......Snip......

I don't think anything about the definition of a game forces someone to try to win. After all in game theory it's well-defined what happens for all possible moves, including obviously bad/losing moves.
It's more of a meta social contract that most of us of course find the game more meaningful and fun.....
I'm not mixing game and meta-game, but actually trying to keep them as separated as possible . That's why I emphasized all the in game and out of the game in my post.
By choosing two-players, deterministic games and trying to avoid "real life" metagame goals, I strive to play really "perfect information" abstract games.

The definition of a game in game theory (taken from Wikipedia) is

"A set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies"

The payoff matrix is integral part of the game. If metagame goals change the real payoff matrix (for example the opponent has a 1000$ prize to lose), we still have a game.
However the new game will not be perfect information because I don't know the real goal of my opponent (which is to lose). Or even if I know it, the game will not be zero-sum because one outcome (my win) is beneficial to both players.

Quote:
Hmm. The masochist who liked a cold shower in the morning so took a hot one.
Despite the valiant attempts at making this complicated, it's actually disappointingly straightforward. The preferred outcome for both players may be my win, but the best (as defined by the terms of the game) is still in each player's case a win for himself.
Correct. "As defined by the terms of the game" excludes any metagame influence. But if you allow metagame goals to change the payoff matrix (which is part of the terms of the game) you are effectively changing the game itself.
It is still a game, but will not be perfect-information or zero sum.
The preferred outcome for a player has to be the best for himself, so I don't see a distinction.

All this, not for attempting to complicate any matter (but thanks for the "valiant") .
I'm just saying that if we allow meta-game to influence the game itself (like the "Nature" player which spurred the discussion) then we must update the definition of the game, for example:

- From two player to three player (if I include "Nature")
- From perfect-information to non perfect information (if I don't know the payoff of my opponent)
- From zero-sum to collaborative (if the max payoff of my opponent is the same as mine).

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megamau wrote:
I'm not mixing game and meta-game, but actually trying to keep them as separated as possible . That's why I emphasized all the in game and out of the game in my post.
By choosing two-players, deterministic games and trying to avoid "real life" metagame goals, I strive to play really "perfect information" abstract games.

The definition of a game in game theory (taken from Wikipedia) is

"A set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies"

The payoff matrix is integral part of the game. If metagame goals change the real payoff matrix (for example the opponent has a 1000$ prize to lose), we still have a game.
However the new game will not be perfect information because I don't know the real goal of my opponent (which is to lose). Or even if I know it, the game will not be zero-sum because one outcome (my win) is beneficial to both players.
This still makes no sense to me. (And I can't figure out if we're having a mere language confusion, or a deeper disagreement of what a "game" is or something...) The real-life (meta-game) $1000 has nothing to do with the in-game definition of who wins a game of chess.

Imagine we are playing a game of chess as people normally would, both of us trying to win as chess players typically do, and then someone watching tells you "If you win, I'll give you $1." Now your meta-game payoff has changed, but surely you agree we're still playing the same game of chess... right? And if a second bystander tells you "Yeah, and if you win, I'll give you $1 also!", surely we're still playing the same game of chess, even though you have some external motivation to try (a little) harder to win. If our external motivations (which in real life are probably continually changing) alter "the game" we are playing, then it seems like we could never claim we are simply "playing chess", but that rather one second we're playing a certain chess variant where I have no monetary motivation, then a minute later we're playing some other chess variant where some money has altered my motivation, maybe a minute later I realize I'm really hungry and sleepy and my motivation to win drops, so again we're playing some new chess game with a different payoff matrix, and maybe a minute later I get a phone from my wife upset about some friend's work crisis and now we're playing yet another chess variant where my motivation has dropped because I'm distracted and worried about some people's problems, and then a minute later... To me, this seems an unnatural and useless way to look at the chess game itself - which is the same game the entire time - same rules, same end condition, same definition of who wins. All those external motivations and payoff matrix changes are about the meta-game, not about the chess game itself. In the chess game itself, there's only "you win, or I win, or it's a draw.".

Quote:
Correct. "As defined by the terms of the game" excludes any metagame influence. But if you allow metagame goals to change the payoff matrix (which is part of the terms of the game) you are effectively changing the game itself.
It's not changing the game itself. It's only changing the real-world consequences of the game.
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First of all I would like to excuse myself to the original poster for hijacking his thread. I'll try to keep it short, because I understand Russ point and probably we'll have to just agree to disagree.

Quote:
The real-life (meta-game) $1000 has nothing to do with the in-game definition of who wins a game of chess.
Correct, but if the external factor, which "has nothing to do with the game", influences the game itself, then we have to acknowledge the fact that the game has changed.
If I suddenly start to play chess to lose, the game will be drastically changed, and this should be easily recognizable even by just looking at the move list.

Quote:
Imagine we are playing a game of chess as people normally would, both of us trying to win as chess players typically do, and then someone watching tells you "If you win, I'll give you $1." Now your meta-game payoff has changed, but surely you agree we're still playing the same game of chess... right ?
-------SNIP------
If our external motivations (which in real life are probably continually changing) alter "the game" we are playing, then it seems like we could never claim we are simply "playing chess"

As in most things it is a matter of degree. Unfortunately we can never completely "abstract" the game from real life, after all we are all playing in real life. So we have to accept small influences like variations in concentration or effort, health, tournament situation, etc. (e.g when drawing guarantees a prize for both players the result will usually be a quick draw). If you play Magic you will know that cards availability and prices influence player decision on which deck to play, sometime more than strategic consideration.

However at some point the external factor completely modifies the goals of the game: one or both players playing to lose, one player playing randomly, extreme kingmaking for outside factors, etc.
In this cases I merely argue that the classification of the game has to be considered changed. It may not be "zero-sum" or "perfect information" anymore or it may have changed to a different game. After all giveaway checkers is different from standard checkers.

All this, however is quite rare in abstract games. You won't easily find somebody paying you to lose (as in boxing ).
So the real point I was trying to make is about randomness. I don't think games like backgammon can be really called "perfect information" just by the introduction of a fictional "Nature" player. We are just sweeping the issue of imperfect information under the carpet defining them as three player games, with one player always in the "king making" position and playing with no strategy nor payoffs.
This disqualifies them from "true" abstracts in my view.
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Christian K
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I think everyone understands what everyone thinks
 
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christian freeling
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Muemmelmann wrote:
I think everyone understands what everyone thinks
That's the most played game during pleasant social interaction
 
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Matteo Perlini
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Russ, I quote James D. Morrow, from his "Game Theory for Political Scientists":
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Game theory assumes rational behaviour. But what is meant by rationality? [...] Put simply, rational behaviour means choosing the best means to gain a predetermined set of ends. It is an evaluation of the consistency of choices and not of the thought process, of implementation of fixed goals and not of the morality of those goals.

The game theory assumes rational agents, that is each player try to maximize his expected utility considering what he know, usually pay-offs and available strategies.

We should consider that game theory has a technical meaning of the word "game". So, as said by Maurizio, if two chess players play chess with different goals (one player plays to win, the other to lose), the "game" (in the technical meaning) changes. Payoffs contribute to specify a "game", so if you change them you change the game.

The same thing happens if you play alone versus the random player Nature, you change something very important from a technical point of view, and that is no longer a "game".
 
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Benedikt Rosenau
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qwertymartin wrote:
Zickzack wrote:
Question: a pack of cards has been shuffled. What is the chance that the topmost is an ace of spades? Answer: barring Schrödinger's cat, it either is an ace of spades or is not. It is not a question of probabilities. However, if the question were what a fair bet would be, 1/52 is the answer.
Go read up on Bayesian statistics! :)
That is where I got this example. Your point?
 
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Christian K
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Interestingly, I think 'no theme' is the least important of the abstract games (we care more about luckless and perfect information). Just thought it is fun since it is the word we use for these games.
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