Recommend
9 
 Thumb up
 Hide
151 Posts
1 , 2 , 3 , 4 , 5  Next »  [7] | 

Abstract Games» Forums » General

Subject: Measuring "fairness" in finite, drawless, two-player perfect information games of no chance rss

Your Tags: Add tags
Popular Tags: [View All]
Nick Reymann
United States
Barberton
Ohio
flag msg tools
designer
"Doubt is not a pleasant condition, but certainty is absurd." Voltaire
badge
Crafty Shaman Impersonator
Avatar
mbmbmbmbmb
Note: As suggested by the title, the following applies only to finite, drawless, two-player perfect information games of no chance.

I put "fairness" in quotation marks because by Zermelo's theorem, in any such game, one of the two players must have a winning strategy. But is there not a way in which we can say, for example, that Hex on a 11x11 board is more "fair" than Hex on a 5x5 board?

In trying to answer this question, I came up with two measurable qualities that could possibly be used in such a calculation:

1: The existence of a simple algorithmic winning strategy for one player heavily weighs against its fairness rating.
Notes: Because an algorithm can be anything from a simple mirroring strategy to checking the entire game tree, I have to put the "simple" qualifier in to make it not include every game. However, what makes an algorithm "simple"? One possible definition is that an algorithm is "simple" if it is solely dependent on your opponent's previous move, and not of the game-state as a whole. Bridg-It and Nim are two good examples of games with such winning algorithms. I'll have to look into this more.

2: The less a player with a winning strategy can deviate from it in order to not turn the game over to the other player's favor, the more fair it is.
Notes: This comes about with the following thought experiment: suppose we have two games where the first player has 100 different moves available. Then suppose they have a winning strategy in both, but in one of them, they can force a win with 75 of those moves, while in the other, they can only force a win with 10. I tend to think of the latter game as more fair. Generally, games with higher game-complexity measurements tend to do better in this category. Also, most combinatorial games fit perfectly here, but many do not fare well in the non-existence of a simple algorithm department.

Are these two qualities sufficient in measuring fairness? I cannot at the moment think of anything else that would apply to such games. Is there anyone here with a more rigorous background that can enlighten me on this topic?
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
Yeah, it seems to me that "fairness" is an empirical practical thing, since indeed theoretically no such game is "fair".

So I think a relevant question about "fairness" is: in practice, what are the win/lose percentages from a given starting position? If it's close to 50/50 then it's "fair". But if one side tends to win significantly more often, then it's not "fair".
4 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Old Gamer
United Kingdom
flag msg tools
badge
Avatar
mbmbmbmbmb
russ wrote:
Yeah, it seems to me that "fairness" is an empirical practical thing, since indeed theoretically no such game is "fair".

So I think a relevant question about "fairness" is: in practice, what are the win/lose percentages from a given starting position? If it's close to 50/50 then it's "fair". But if one side tends to win significantly more often, then it's not "fair".
This. You can't really have a universally fair game with the conditions you state above, you can only come up with a game that is fair for players with normal human limits...

For small children, Tic-Tac-Toe could be fair, but most adults (and older children) presented with the game for the first time would swiftly solve it.

For humans, chess is a (relatively) fair game (I believe win ratios for evenly matched players are around 55:45 for white?) but for chess playing computers 50 years from now it may be solved - rendering it completely unfair.

Russ's notion of fairness seems the sensible one to use...
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Nick Reymann
United States
Barberton
Ohio
flag msg tools
designer
"Doubt is not a pleasant condition, but certainty is absurd." Voltaire
badge
Crafty Shaman Impersonator
Avatar
mbmbmbmbmb
The problem I have with that definition of fairness is that it is completely dependent on the skill levels of those that are playing. Take Connect 4 for example: the better two players are, the more likely the first player will win. I'm looking for more objective qualities.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
CB

Tennessee
msg tools
But fairness isn't objective.

In Go, the handicap system is dependent on the opposing skill levels of the players. Player Rank and game outcome are dependent on each other. Your rank is stable when you have 50/50 win/loss ratio against players of the same rank.

Likewise, the evolution of Komi (a balancing mechanism) is interesting. It is not set but evolves and changes of the history of the game. Check out Senseis on Komi .. especially History of Komi and "Fair Komi" (as opposed to Perfect Komi) subsections. http://senseis.xmp.net/?Komi
1 
 Thumb up
0.02
 tip
 Hide
  • [+] Dice rolls
Old Gamer
United Kingdom
flag msg tools
badge
Avatar
mbmbmbmbmb
Mingy Jongo wrote:
The problem I have with that definition of fairness is that it is completely dependent on the skill levels of those that are playing. Take Connect 4 for example: the better two players are, the more likely the first player will win. I'm looking for more objective qualities.
Yes, and Connect 4 is a (moderately) fair game for low skill players, degenerating to a completely unfair game for skilled players.

Or do you propose to apply a single measure of fairness to Connect 4 (and every other finite, drawless, etc... game (henceforth FDTPPIGoNC)) despite this? What would it mean to call the game unfair (or give it a low fairness value) when low skill players get results which don't depend (much) on starting player? Or to call it highly fair when it is solved for high skill players?

 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
Mingy Jongo wrote:
The problem I have with that definition of fairness is that it is completely dependent on the skill levels of those that are playing. Take Connect 4 for example: the better two players are, the more likely the first player will win. I'm looking for more objective qualities.

But if the game is "fair", then that difference between the stats for weak players and the stats for strong players will be small.

If, between strong players, the first player tends to win a lot with Connect 4, then that seems to me to indicate that the game is not "fair". I.e. the definition "works", it catches that Connect 4 is not very fair. So I'm not sure I see the problem.

If your point is that this definition of fairness requires you to use fairly strong players to assess the fairness, then yes, I agree, but I don't see how that's a problem. If weak players of similar strength play, then sure their game results are going to be more random/chaotic, even if the game is not very fair. I.e. "fairness" isn't even very relevant for weak players. (Unless it's blatantly unfair - but then you'll see it even with weak players.)
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Steven Backues
United States
Ann Arbor
Michigan
flag msg tools
mbmb
russ wrote:
Mingy Jongo wrote:
The problem I have with that definition of fairness is that it is completely dependent on the skill levels of those that are playing. Take Connect 4 for example: the better two players are, the more likely the first player will win. I'm looking for more objective qualities.

But if the game is "fair", then that difference between the stats for weak players and the stats for strong players will be small.

If, between strong players, the first player tends to win a lot with Connect 4, then that seems to me to indicate that the game is not "fair". I.e. the definition "works", it catches that Connect 4 is not very fair. So I'm not sure I see the problem.

If your point is that this definition of fairness requires you to use fairly strong players to assess the fairness, then yes, I agree, but I don't see how that's a problem. If weak players of similar strength play, then sure their game results are going to be more random/chaotic, even if the game is not very fair. I.e. "fairness" isn't even very relevant for weak players. (Unless it's blatantly unfair - but then you'll see it even with weak players.)


Taking this further... One empirical measure of "fairness" could be, "how strong do the players have to be before the win percentages deviate from 50/50?"
5 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Joe Joyce
United States
Yonkers
New York
flag msg tools
designer
Avatar
mbmbmbmbmb
Might I suggest that a very high level of complexity relative to any potential players' abilities and capabilities would be necessary to guarantee "fairness". There is a game called Wormhole Chess, where squares that a moving piece starts its move from are removed from the board, and have collapsed into wormholes, which act to directly connect the first two remaining squares on opposite sides of the wormhole. If squares 2 and 3 in a row are gone, square 1 connects directly with square 4. Needless to say, the game can't go beyond 32 moves, making it very finite. The game pieces are not all the standard chess pieces, but some new pieces suited to the wormhole board. You must legally move a piece each turn, or lose, as well as losing by checkmate. I don't believe there is any possibility of a draw. But the game is so complex that it is "fair" in the general sense of the word.

 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Joe Joyce
United States
Yonkers
New York
flag msg tools
designer
Avatar
mbmbmbmbmb
Russ, you said this:"But if the game is "fair", then that difference between the stats for weak players and the stats for strong players will be small."

Permit me to respectfully disagree here. In my previous post here, I discussed Wormhole Chess. It seems to me that a good player could pretty consistently beat a mediocre player, regardless of which sides they took in any game. And I would consider that a test of fairness of an abstract strategy game in general, whether or not it is drawless, that the better player wins appropriately more often. To me, if good and poor players had an equal chance, the game would be random. Are we looking at different things, or am I misunderstanding something? Because I honestly believe that to be fair, a game must allow draws and also provide the better player a better chance of winning. Skill vs skill, essentially. Otherwise, to me, it seems to be an interactive puzzle.

And that leads me to the original post, but maybe in another comment.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Joe Joyce
United States
Yonkers
New York
flag msg tools
designer
Avatar
mbmbmbmbmb
2097 wrote:
joejoyce wrote:
Might I suggest that a very high level of complexity[…]

I’m not too fond of very opaque games.

Go is in a weird sweet spot for me where it’s pretty clear what you’re supposed to do while at the same time there is much that you can learn.
Other games seem to have either a “yeah, obvious, my old dosbox could solve this in 10 minutes with BASIC” or a “what the heck am I supposed to do? what’s a ‘good position’?”. Seems like wormhole chess would fall in the latter category.


It's been a few years since I played it, but it is a very nice game, and if you do enjoy chess variants [which I admit is a highly specialized field] that is an excellent one. I found it complex but not opaque. It does take a little practice to get used to, but you can see strategies and tactics. It's just that the surprise factor is higher, and the board is shrinking, limiting your options each time a piece moves. You can get caught badly by the limits, literally running out of board in which to dodge. For one thing, you are absolutely in the end game by turn 20, because there are only a couple dozen board squares left at that point.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Joe Joyce
United States
Yonkers
New York
flag msg tools
designer
Avatar
mbmbmbmbmb
Item 2 from the original post says:"2: The less a player with a winning strategy can deviate from it in order to not turn the game over to the other player's favor, the more fair it is."

Again, I am misunderstanding maybe what you are saying, but I see your ideal game as a puzzle to be solved, rather than a game where chances are equal on each side. Grin, in practice, there may be no visible difference between the two. But in theory...

To exaggerate the point considerably, in theory, your puzzle games are all sort of tic-tac-toe, except that there is a guaranteed winner, rather than a guaranteed draw. And I've just been called to start dinner, so I'll come back for my abuse later. Enjoy! :)
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Brian M
United States
Thornton
Colorado
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
Quote:
Russ, you said this:"But if the game is "fair", then that difference between the stats for weak players and the stats for strong players will be small."

Permit me to respectfully disagree here.

I believe Russ was referring to weak players vs. weak players and strong players vs. strong players, not weak vs. strong players.
4 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Joe Joyce
United States
Yonkers
New York
flag msg tools
designer
Avatar
mbmbmbmbmb
StormKnight wrote:
Quote:
Russ, you said this:"But if the game is "fair", then that difference between the stats for weak players and the stats for strong players will be small."

Permit me to respectfully disagree here.

I believe Russ was referring to weak players vs. weak players and strong players vs. strong players, not weak vs. strong players.


Thanks, StormKnight. I should probably not post with temp and a raging headache. Russ doesn't say things I generally disagree with very often. Seemed like a good idea at the time... permit me to apologize, then. [What really hurts is when I make tournament moves in that condition. I've had some ignoble losses...]
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
joejoyce wrote:
Russ, you said this:"But if the game is "fair", then that difference between the stats for weak players and the stats for strong players will be small."

Permit me to respectfully disagree here. In my previous post here, I discussed Wormhole Chess. It seems to me that a good player could pretty consistently beat a mediocre player, regardless of which sides they took in any game.

Yes, as StormKnight hypothesized, you indeed misunderstood what I meant, because I probably didn't express myself sufficiently clearly.

Of course I agree that a stronger player will tend to beat a weaker player in a game of skill.

What I meant in the quoted text was this:

If you look at the statistics from a lot of games between weak players to see if there's a first or second player advantage, and you look at the statistics from a lot of games between strong players to see if there's a first or second player advantage, then both those analyses will show close to 50/50 wins for first/second player if the game is "fair".
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
2097 wrote:
I’m going to go in a different direction.

A deterministic game is fair as long as it is not known to the general public which player has the advantage.

Simply knowing who has the advantage seems insufficient to call a game "unfair".

E.g. Hex with the pie rule (heck, any game with the pie rule) is known to have a second player advantage. But I don't think that makes Hex unfair. (Do you really think Hex is unfair? Or am I misunderstanding what you mean?)

Quote:
So for me, solving a game means ruining it, in one sense. Still interesting to do though, so by all means, don’t stop, game solvers!

For me solving a game has no particular implications for whether it is "ruined".

Hex is weakly solved (a known win for the second player), yet it's not ruined.

Checkers is even strongly solved (in the full sense of proven known best moves from all possible positions), yet it's not ruined since no human player can practically carry out the winning moves with certain reliability - it's too complex.

Discovering a simple human-usable strategy guaranteed for a given side to win is what "ruins" a game, I would say. E.g. Tic-tac-toe.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
2097 wrote:
I’m not so sure. In some tafl games one side has an advantage in weak vs weak, and the other side a supposed advantage in strong vs strong.
Fascinating.

Oh, that's a good point for very asymmetric games. A classic non-abstract example is Mr. Jack (where many newbies post in the forum claiming that it's impossible for the killer to win... or sometimes claiming that it's impossible for the detective to win...).

Quote:
If the game is 50/50 between strong players it’s fair, why bring in weaker players in the mix? Having the game be unbalanced when your new is no problem I think. You can have the weaker player take the “easier” side.

Yep, I agree. Ultimately fairness matters for stronger players, not newbies.

But from a "marketing" point of view, it can be a problem if new players erroneously believe a game is horribly unfairly unbalanced. A wise publisher might include some note in the rules that new players often find side X easier to win with than side Y, but that this difference disappears as players gain insight and experience.
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
2097 wrote:
I don’t know much about Mr. Jack because I don’t like killing games

For what it's worth, no killing occurs in the game. It's actually a deduction/maneuver sort of game.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
David Bush
United States
Radiant
Virginia
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
Mingy Jongo wrote:

I put "fairness" in quotation marks because by Zermelo's theorem, in any such game, one of the two players must have a winning strategy. But is there not a way in which we can say, for example, that Hex on a 11x11 board is more "fair" than Hex on a 5x5 board?

Well for Hex specifically, sure. Computers have worked out a swap map for Hex up to 8x8 and for many initial moves on an 9x9 grid, as well as the exact way to win for the second player in all mapped cases. No it's not simple like Brigit, but strong players generally prefer larger grids.

Quote:
...

2: The less a player with a winning strategy can deviate from it in order to not turn the game over to the other player's favor, the more fair it is.
Notes: This comes about with the following thought experiment: suppose we have two games where the first player has 100 different moves available. Then suppose they have a winning strategy in both, but in one of them, they can force a win with 75 of those moves, while in the other, they can only force a win with 10. I tend to think of the latter game as more fair.

It sounds as though you refer to not just two specific played games, but two specific positions during the course of these games. Do you mean, in the opening positions of each game? It would likely be impossible to determine such values unless the game were solved, in which case your first criterion would apply. Anyway, IMO the number of winning moves available is less relevant than, although not entirely unrelated to, the difficulty of finding a winning plan.

I agree with Russ that the statistical results from played games provide the best practical measure of fairness. The more fair a game is, the deeper it is. ELO ratings in particular can tell you a lot about how deep a game can go, although they don't prove anything.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
George Leach
United Kingdom
Godalming
Surrey
flag msg tools
designer
Avatar
I'd rank levels of fairness as follows:
- Plays within 5% of 50/50 at all levels of play.
- Plays within 5% of 50/50 at the higher levels of play.
- Plays within 10% of 50/50 at the higher levels of play.
...
- Plays within 25% of 50/50 at the higher levels of play.
- Is weakly solved by human players.
- Is strongly solved by human players (i.e. tic-tac-toe).


This unfortunately means that games will vary along this scale depending on the top level of play. New games my seem unfair until well studied.
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
Jugular wrote:
I'd rank levels of fairness as follows:
- Plays within 5% of 50/50 at all levels of play.
- Plays within 5% of 50/50 at the higher levels of play.
- Plays within 10% of 50/50 at the higher levels of play.
...
- Plays within 25% of 50/50 at the higher levels of play.
- Is weakly solved by human players.
- Is strongly solved by human players (i.e. tic-tac-toe).

What do you mean by "weakly solved by human players" (perhaps give some examples) and "strongly solved by human players"?

I guess the latter means that humans can in practice play the game optimally. I agree that's an extreme end of the fairness spectrum.

But I'm not sure what you mean by "weakly solved by human players", and it sounds potentially fishy in a fairness spectrum.
1 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
George Leach
United Kingdom
Godalming
Surrey
flag msg tools
designer
Avatar
russ wrote:
Jugular wrote:
I'd rank levels of fairness as follows:
- Plays within 5% of 50/50 at all levels of play.
- Plays within 5% of 50/50 at the higher levels of play.
- Plays within 10% of 50/50 at the higher levels of play.
...
- Plays within 25% of 50/50 at the higher levels of play.
- Is weakly solved by human players.
- Is strongly solved by human players (i.e. tic-tac-toe).

What do you mean by "weakly solved by human players" (perhaps give some examples) and "strongly solved by human players"?

I guess the latter means that humans can in practice play the game optimally. I agree that's an extreme end of the fairness spectrum.

But I'm not sure what you mean by "weakly solved by human players", and it sounds potentially fishy in a fairness spectrum.



Sandra's wikipedia quote sums it up as I was intending it. Weakly or Strongly solved and such solution is applicable by humans in normal play. So to be weakly solved by human players would require that there's some heuristic/algorithm that ensures a win that can be applied by human players. This would require that the algorithm be easily calculable within normal game time limits. I believe and example would be Bridg-it,
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Christian K
msg tools
Avatar
mbmbmbmbmb
Sandra, that is an interesting post. I hope this will not take your joy away from games forever, but you could argue for any game that can be solved (ie. all full information games) the winning strategy exists somewhere. Or is it okay if noone knows it (computer or human)
 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Stephen Tavener
United Kingdom
London
England
flag msg tools
designer
The overtext below is true.
badge
The overtext above is false.
Avatar
mbmbmbmbmb
2097 wrote:
Yes, as soon as the computer finds out, it changes things, but if neither computer nor human knows, it’s still fair.
Look earlier in this thread, I wrote:
2097 wrote:
A deterministic game is fair as long as it is not known to the general public which player has the advantage.


So, how do you feel about the swap rule? The swapper always has the advantage, since they can play on if the game is in their favour, or swap otherwise. Of course, they may not know whether a given opening favours them or not...
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
Russ Williams
Poland
Wrocław
Dolny Śląsk
flag msg tools
designer
badge
Avatar
mbmbmbmbmb
2097 wrote:
Jugular wrote:
Sandra's wikipedia quote sums it up as I was intending it.

Good, I guess understood you correctly. So if a reasonable expectectation of what a human is can apply these algorithms in typical play situations, the game is not fair.

Personally, I disagree. To my own tastes though, as soon as the game is weakly solved by some supercomputers, I perceive it as way less fair because you now know for sure that one side does have an advantage.

But we already know that for every combinatorial game which doesn't permit ties, one player or the other has a guaranteed win if they play optimally. That's a basic result of combinatorial game theory.
2 
 Thumb up
 tip
 Hide
  • [+] Dice rolls
1 , 2 , 3 , 4 , 5  Next »  [7] | 
Front Page | Welcome | Contact | Privacy Policy | Terms of Service | Advertise | Support BGG | Feeds RSS
Geekdo, BoardGameGeek, the Geekdo logo, and the BoardGameGeek logo are trademarks of BoardGameGeek, LLC.