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We did it! For the first time, Manifold won a game of Manifold plays chess. Thank you for participating!

Things that I might do different for next time:

- resolve conditional markets to main market (average over some pre-determined times)

- probably don't do the move penalty for score

- only create 2 conditional markets each move - not clear that 3 markets were better

- do the average over the last hour instead of last 4 hours

- use some mechanism that the worse conditional market has a chance to resolve. like 1% or 2% probability the losing move gets chosen. But bigger chance if the conditional market outcomes are very close to each other.

However, I will not run Manifold plays Chess 4 anytime soon (and I decided on that before I lost all my mana if you ask). If others want to run that, or run another game, now might be a good time. Maybe I can even help you in some way.

@harfe Thanks for running it. I appreciate that it was a lot of effort and mana. :)

@harfe Thanks, and congrats!

@harfe Thanks for running the game! I had fun seeing how the game and the market mechanism played out and manipulating them for some profit. :)

https://manifold.markets/harfe/manifold-plays-chess-3-if-we-play-4-b9b5d11507a2

potentially last market

https://manifold.markets/harfe/manifold-plays-chess-3-move-42-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-41-what-c06064831380 we could win very soon

https://manifold.markets/harfe/manifold-plays-chess-3-move-40-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-39-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-38-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-37-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-36-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-35-what

can the latest moves be added here? thanks

https://manifold.markets/harfe/manifold-plays-chess-3-move-34-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-33-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-31-what

Wow this market is optimistic! Please predict here:

https://manifold.markets/harfe/manifold-plays-chess-3-move-30-what

https://manifold.markets/harfe/manifold-plays-chess-3-move-29-what pls predict and/or choose moves

https://manifold.markets/harfe/manifold-plays-chess-3-move-28-what Our opponent has only 5 pawns and a king left. How should we continue?

I am considering testing "leveraging", as mentioned in the "possible rule change during the game". Originally the idea is due to @citrinitas (I forgot where). See also some discussion here.

Why? The score of the conditional markets are stagnating, and profits are low. The game looks to be relatively calm. Also I would see this as an experiment, rather than a thing that improves accuracy.

Downsides: This would mean additional complexity and would make it more complicated to understand for participants how the mechanism works in detail.

@harfe as for the concrete transformation formula, I am not sure. One considerations is to have a market value of 0.1 correspond to a score of 0.7, and linearly interpolate for the rest (with a market value of 0.0 corresponding to a score of 0.0 and 1.0 corresponding to 1.0). With the additional rule that we drop back to un-leveraged markets once the score goes below 0.7 (upon resolution).

@harfe I think the most manipulation-resistant resting point for the leverage is at 0.5 -- at either end, it becomes proportionally cheaper for an attacker to shift the market towards 50%. I'm not super confident about that, though.

Most things I've said about manifold chess are collected at https://manifold.markets/post/manifold-plays-chess-2-retrospectiv

@citrinitas interesting argument. That was also an alternative I considered in the discussions with Daniel: choose the transformation so that the previous score corresponds to a probability of 50%. Downside is that the formula changes every move, which might make things even more complicated.

@harfe

>market value of 0.1 correspond to a score of 0.7

In this model a neutral 50% market value would correspond to a score of 0.8333 which I think is too high for a neutral market in the current game position (there’s still many ways to blunder white’s advantage as shown by one of the current move candidates). Making the neutral 50% correspond to a score of 0.7 would be more sensible, imo.

@citrinitas

>the most manipulation-resistant resting point for the leverage is at 0.5

Can you unpack this a bit for me? What is resting point, is it related to the recent score of 0.7-0.75? And are you saying that should correspond to a 0.5 (50%) market, similar to what I suggest above?

@harfe

>Downside is that the formula changes every move

In our discussion you proposed having the same formula and I liked the simplicity, except we’d revert back if the score tanked, like you suggested here. And it can be piecewise linear (though I like me a smooth curve, which btw I think I fixed the error I had made with that first suggestion). You had mentioned something like (0.1,0.9) mkt prob <-> (0.65,0.85) score, and interpolate the rest. What’s the downside of this for all moves (as long as score>.65)?

@deagol Resting point is what I'm calling the value that the conditional markets should resolve to, if there is no change in the evaluation market.

I think the most "mathematically pure" way to work this would be with a log-odds transformation. For example:

import math

Logit = float
Prob = float

def log_odds(p: Prob) -> Logit:
    return math.log(p / (1-p), 2)

def inv_log_odds(l: Logit) -> Prob:
    return 2**l / (1 + 2**l)

def score_market_resolution(eval_prior: Prob, eval_posterior: Prob) -> Prob:
    scale_factor = 1.0
    return inv_log_odds(scale_factor * (log_odds(eval_posterior) - log_odds(eval_prior)))


for prior, posterior in [(0.7, 0.8), (0.7, 0.7), (0.7, 0.4), (0.4, 0.7)]:
    score = score_market_resolution(prior, posterior)
    print(f'{prior} -> {posterior} == score {score}') 

results in

0.7 -> 0.8 == score 0.6315789473684211
0.7 -> 0.7 == score 0.5
0.7 -> 0.4 == score 0.2222222222222223
0.4 -> 0.7 == score 0.7777777777777777

Here raising the 'scale_factor' makes the leverage stronger without disrupting the smoothness of the curve.

This feels like the kind of thing where some Robin-Hanson-esque grad student proves it's optimal for some condition under certain constraints, and then it turns out you violate those constraints in the real world but it performs pretty well anyway.

But I haven't done any of that analysis and I'm just going on vibes, so take this with a grain of salt.

@harfe I think the result is getting close to ordained now, so might as well experiment away!

@citrinitas yeah, it’s pretty much what I was trying to get to here (stumbled by trying to use a generalized logistic which broke the 50% symmetries) but as I mentioned above I think I got it right now. The only difference is I think you swap the concept we’ve been using as ‘score’ with the conditional resolution value. This is what I got for the smooth score as a function of its winning conditional’s last 4 hours average probability p, the previous move score s, and a leverage factor L:

score(p) = 1 / ( 1/odds(s)/odds(p)^(1/L) + 1 )

I can’t prove that’s mathematically pure or optimal, just, same as you it seemed sensible to multiply the odds of the conditional (scaled) by the odds of the previous score, then invert the logit. But then @harfe prefers the piecewise linear, which I don’t mind.

One reason why I like piecewise linear: linearity is nice because it plays well together with expected value. piecewise linear also does break linearity, but at least it is locally preserved.

Another argument against leveraging that just came to mind: if the market values are more extreme at close, it means a higher portion of the subsidy gets in the pocket of traders. And currently at near 80% it is more extreme than it might be with leveraging. But its a cost I can live with.

Ok preliminary decision now:
Starting move 31 I will introduce a piecewise linear transformation function to convert market values into scores (I called this leveraging originally. Not sure if this is a good name actually).

The market will be centered around 50%, that is if z is the (possibly rounded) score of the previous move, z will correspond to a market value of 0.5.
Other score-market interpolation points are (z - 0.08, 0.1) and (z+0.08, 0.9). If z>0.9, then we use transformation as if z=0.9 (similar rule if z<0.1). We will automatically return back to normal when the difference between consecutive scores is greater than 0.08 twice in a row. I reserve the right to switch to a different formula or to the normal way but will announce this 1-2 moves in advance.

@harfe “We will automatically return back to normal when the difference between consecutive scores is greater than 0.08 twice in a row.”

I’m not following this bit, can you explain?

@deagol ah I guess this is just on a falling score, ok got it