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What is optimal robber placement in Settlers?
Every time I think I've mastered Settlers, I discover yet another novel tactic or strategy. This time it's Robber placement.
I've usually selected robber placement based on a combination of factors, including who is winning, which hex reduces the most dots worth of production (x2 for cities) that isn't mine, several variations on which resources are scarce, whether a 2:1 port for an abundant resource is occupied, and trying to take a card I need from a player I think likely to have it. Which of these factors to emphasize for any given robber placement is a fairly complicated decision that depends on what stage of the game you're in, how much the leader is winning by, and what you're needing next.
All this is standard stuff for those who have played more than a few games. But I just ran across an idea I hadn't heard before:
When not trying to stop someone close to winning, nearly always place the robber on the most valuable hex of the person on your right only. The reason for this is that this person won't be able to retaliate for at least 3 rolls of the dice (in a 4 player game), while the other players may be more favorably disposed to you for not getting them. And if this player (to your right) retaliates, it is immediately your turn next giving you a chance to possibly roll a 7 or play the robber (if you have it). So assuming robber retaliation is somewhat predictable, this seems to be a sound strategy.
I don't have a strong opinion yet on how well the person-to-the-right robber placement strategy works - as I don't know how predictable retaliation is in general or with the specific groups I play with. But I do now wonder:
For advanced Settlers players, what is optimal robber placement? Is it this player-to-the-right placement method (because retaliation is mostly predictable)? Is it the standard tactics I listed above? Or are there different and perhaps more subtle tactics for Robber placement I haven't even thought of?

Am I the only one to think that if you are going so deep in strategical and tactical aspects of a game like Settlers, then maybe you should play another game? I mean: Settlers is a family game, usually played for fun, where luck is a strong aspect of the game. There are a lot of games where thinking deeply on strategy and tactic will be worth the effort, but not Settlers.

@Djaian I don't think you're the only one who thinks that way. However, despite the seemingly big luck factors in games like Settlers, Dominion, Stone Age, etc. it is interesting that the very best players win most of the time and consistently do well in tournament settings. Perhaps my favorite of all games is the card game of Bridge which is obviously a fairly deep game. That doesn't mean I can't explore Settlers to its strategic limits, which apparently I haven't yet found.

I can't figure out if you are asking one overly broad question, or many smaller-scoped questions... in any case:
The "player to the right" strategy ignores a few possibilities, including:
 - one or both of the two players after you roll a seven,
    - one or both of the two players after you play a soldier,
    -  your victim is known to (not) have in a particular resource in hand.
Retaliation is predictable in the sense that good players will make decisions based on cost/benefit/probability analysis (including "dots" on the chits, scarcity of resources, etc.), while poor players will be swayed by subjective reasoning (retaliation, etc.).
We play with promises of immunity, bribes -  you name it. This makes it harder to pin-down a specific strategy, but I will say that it's mostly the analytical approach that you mention.

My answer is very subjective:
I play for fun and not very competitively so when placing the robber, I take into account the "human factor". This is my thinking, "where you place the robber isn't as important as who you hurt by placing the robber". Maybe I just play with really vindictive players, but I find the robber really pees some people off. They stop trading with you and the talk trash about you, turning other players against you and now you have no chance of non-bank trading. (Also in the Cities and Knights expansion, people can really hurt you if they want to.)
The best Catan player I know places the robber on a 2 then 3,4,5,6,8,9,10,11,12 just so people know he's not trying to hurt anyone. It seemed to work for him.
I try to play on the 6 or 8 that two other players have built around. That way at least your spreading the pain around rather than targeting an individual.

+1 for stressing the importance of the human factor.  If you always play the robber to maximise the amount of pain, you aren't going to be a popular guy at the table.  Personally I usually try to target the person who is (or is perceived to be) in the lead; or if I'm in the lead, the other player who is giving me the most grief about that!  Playing the political game seems more important than the marginal utility of occasionally depriving a player of resources, unless maybe that player will win the next time a 6 is rolled or whatever...

For the reasons you mentioned, you should place the robber on the player to your right, ALL OTHER THINGS BEING EQUAL.
Usually they aren't. But your "rule" is a good "tiebreaker." Meanaing that it is one of SEVERAL considerations (you've named most of the others).

As you say, the decision of where to place the robber depends on a lot of factors, and their weight depends on the particular situation. I think it is worth considering turn order and likely robber residence-times, but these are just additional factors to consider. And whether each player has unplayed development cards that could be soldiers should be considered along with seat placement. If everyone has (potential) soldiers, then, all else being equal, I think the player-to-the-right would be the optimal choice.

For advanced Catan players this is a non-topic as retaliation itself is a non-topic. Why should I block you with the robber if it doesn't benefit me the most or why should I not block you if it benefits me the most?
What you should weigh againt each other are things like "what do I get" vs. "what does the person blocked lose" as it might be more beneficial to get a much needed resource than having someone else not get something or the other way around.
Blocking a resource so others don't get it while you still get a bunch WOULD give you a monopoly on it and a favorable trade position, BUT it would only really be worth it in the long run and a 7 is statistically speaking coming up every 6 rolls, which is hardly enough time to get the others into a tight spot and force them to trade with you in expensive trades.
Both things I mentioned might be tie breakers if everything else is the same...but that's very very rarely the case to begin with.

retaliation can be a strong tool if you use it right: if you are known to be the retaliating kind it virtually costs more to attack you. The key is to make it the most believable possible that you will retaliate while actually retaliating the less possible.

It's just a weak bluff that you either can't go through with or lose with it. If you are in an advaned setting I'll call your bluff every time, because either you don't belong in that setting or you are bluffing.
It's like going all in in poker with a weak hand. Works very rarely (in an advanced setting. I'm not talking about playing your family or friends who don't know their chances to begin with) and when it doesn't, the outcome is horrible.

In poker if someone have a tendency to raise the bid with a weaker hand than what would be optimal, then a clever opponent would take it into account and use this to his advantage, he wouldn't just continue to apply his generic strategy, especially if there are other players. The same goes for Catan: if I have to choose between crossing someone who will hold the grudge for the rest of the game and someone who won't take it into account for his calculations, I will definitely choose to cross the second one.

So incidentally the grudger one will get that small advantage other the calculating one. Of course if he spends some games focusing the first one who offended him he can get a disadvantage from that (depending on the amount of resources he spend it can go from nothing to sacrifice all this chances of winning), but in the long run it tends to pay off.

No, he'll just do stupid attacks that don't benefit him all the time and in the long run everyone will see that it's stupid and he loses all the time with this.

Well, guess you just never played against this strategy. I regularly win with this, against all kinds of players, and the more the players are "advanced" as you say, the best it becomes.

Can you tell me some tournaments you won so I can look you up?


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In Pandemic, do certain setup and card order combinations make it impossible to win?
My family has won most of the Pandemic (2 or 3 player games) we've played, but we ran into one recently that I thought impossible to win due to the 2 and 3 cube infections being far from Atlanta, far from each other, and not accessible with any of the cards we were dealt or received in the first few turns. We lost very early before we obtained even a single cure.
It would have been great if I could have recorded the exact initial setup and card sequence to see if a win was possible - but I didn't.
Has anyone done the work necessary to answer the question: Do certain setup and card order combinations make it impossible to win?

Yes - it's possible not to be able to win.  In fact it's possible to not be able to survive until the second turn.
Here's an example of someone playing a game which they lost on the first turn: http://boardgamegeek.com/thread/373797/it-takes-talent-to-lose-on-the-first-turn
Maybe they could have done better, but if they had cards such that they couldn't treat anything in the black area then that would be the end of the game.  Here's a full example:
Initial placement of cubes are three in Baghdad, Riyadh and Karachi, two in Cairo, Tehran and Delhi and one in Algiers, Istanbul and Moscow (i.e. 18 of the 24 black cubes are already on the board).
Position after initial infection

Four players, player 1 is Medic with Santiago and Beijing. The other players (Operations Expert, Scientist and Dispatcher) have no special events, and don't have Atlanta.  The first player can't treat any cities in four actions, and picks up one city and one epidemic at the end of their go.
The epidemic city is Mumbai, the marker is moved up one space, and after shuffling the city cards Baghdad and Riyadh are drawn.  These cause two large chain reactions with 18 outbreaks and requiring 45 cubes on the board - a loss by two methods.
Position after epidemic in Mumbai, and infection in Baghdad (outbreaks highlighted in green)

Position after infection in Riyadh

Pandemic can be a frustrating game at times.  I think the author of the session review sums this up pretty well:

All that set-up, for less than a minute of game play!

While the accepted answer shows that it definitely is possible, it seems highly unlikely to encounter a situation where you can prove that no win condition is possible.
"I thought impossible to win due to the 2 and 3 cube infections being far from Atlanta, far from each other, and not accessible with any of the cards we were dealt or received in the first few turns."
It should be possible to reach just about any city with any set of cards in 2 turns.  You just may need a little creativity.  Remember that reaching any city that you have the card for allows you to go anywhere.  Want to get to red but only have 2 blacks?  Play one, walk to the other, and then use it to fly to red.  Expensive, yes, but certainly not impossible.
And while having the cities with lots of cubes spread out makes them difficult to remove, it also makes them unlikely to cause chain reactions.  So you don't have to drop everything to try to get to them right away.  You can still win with 7 outbreaks, consider them a resource and "use" every one you can to get you closer to cures.

I thought this was a terrific additional answer as it suggested ways to avoid losing that many people might not think of. I also agree with you that for many setups it would be difficult to prove manually whether or not it is impossible to win. However, computer simulation might be able to provide such proof, if it's not too computationally expensive to explore every possible set of player choices.

This question actually has the answer to how likely it is to lose on the first turn
https://boardgames.stackexchange.com/questions/8487/what-are-the-chances-of-losing-pandemic-on-the-first-turn?noredirect=1&lq=1

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Are 2 2-card-swishes a valid 4-card-swish?
I played my first game of Swish this evening and the other player was surprised when I asked if 2 sets of 2 card swishes are a valid 4 card swish. That is - can you pick up 2 pairs of cards that each are 2 card swishes and stack them together into a 4 card swish, so long as every ball swishes into a hoop of the same color and there is no overlapping as you visually see 4 balls in 4 hoops?
The rules don't specifically comment on this, so I am assuming yes. But given the other player's surprise, I wanted to make sure.
Of course, the same question can be extended to 2+3=5, 2+2+2=6, etc. I asked the question with 2+2=4 in order to provide a specific example.

I decided to email support at Thinkfun.com with:

I played Swish last night for the first time and was unclear about
  something after both reading the rules multiple times and viewing your
  video:
The question is whether a pair of 2-card-swishes constitute a valid
  4-card-swish, so long as the hoops and balls don't overlap among all 4
  cards. The rules don't comment on this situation one way or another so
  I am assuming the answer is yes (and can be extended equally to 2+3=5,
  2+2+2=6, etc., again so long as the hoops and balls don't overlap when
  you stack the subsets together.
Another way of asking the same question is if you have what seems to
  be a valid 4 card swish per the rules, is it still valid if the 4
  cards can be rearranged into 2 sets of 2 swishes?
I asked this question on Stack Exchange (Board and Card Games) and
  the person who answered came to the same conclusion I did - that the
  rules as literally stated indicate this is a valid swish (see link
  beneath my signature). However, this sits uneasily with me because
  2+2=4 swishes are easier to find than 4-swishes that can't be
  decomposed into 2 sets of 2. Yet the scoring system does not reward
  finding more difficult swishes.
If you don't know the answer, can you pass this on the game creators
  to find out their intent?

Here's the answer I received:

Below is the answer to your questions from one of our Production Team
  Members:
The short answer for his question is no, Two 2-card Swishes do not
  “count” as a 4-card Swish. But, they do count as a legal move in the
  game, and the person who finds two 2-card Swishes still wins 4 cards,
  just as many as a person who makes a 4-card Swish. So, to answer the
  second point, at first blush this would seem not to make sense, since
  as Joe points out, the scoring system does not reward the person who
  found the more complicated Swish. This is actually done deliberately.
  The reason is that the winner is simply the player who has won the
  most cards. We tested versions where harder Swishes (higher counts)
  where worth more, but we decided on this version because it allows
  multiple players, of multiple skill levels, to play an equally
  challenging game. As in, when families play, Mom or Dad can be limited
  to 3 or higher card Swishes, whereas Jr.(s) can go for any number.
  This way, all players will be challenged, and Jr.(s) have an equal
  chance to win the game.
We understand that for more experienced gamers, this will seem
  somewhat “off”. But this was the version that had the greater appeal,
  and we assume that people that really get into games like this can
  adjust the rules as they see fit, adding additional points for higher
  card Swish. I would suggest +1 for a 3-card Swish, +2 for 4-card, and
  so on. Keep in mind, though, that if you play this way, you have to
  keep all your Swishes separate as you win them, or else do the math in
  your head and remember it. This is another minor detail that can add
  complication, something that we try to avoid.

I interpret the answer as saying it doesn't count as a 4 card swish yet you can collect 2 2-card swishes simultaneously. It's phrased this way so that if an experienced adult is playing a kid using the variant that kids can do 2-card swishes while adults can't, then the adult is not allowed to simultaneously pick up 2 pairs for a 4 card swish, even if there is no ball/hoop overlap.

Yes, but I agree that the examples aren't completely clear on this matter. It two 2 card Swishes could be overlaid into a 4 card Swish, it doesn't break the rules for "What a Swish is". The examples in the rulebook, and in the video are also no help in this matter, because the rulebook is too brief, and the video only shows examples of 3, 4, and 5 card Swishes being made from cards that need all cards in the set to make the Swish (which wouldn't be the case with a 2+2 Swish = 4 Swish, or 2+3 Swishes = 5 Swish. It is possible this isn't the designers intent, but based upon the rules as written and the fact that an FAQ does not exist that clarifies this question, I believe you should be allowed to claim a 4 card Swish from two independent 2 Card Swishes that can be laid on top of each other.

WHAT IS A SWISH?
A Swish is created by layering two or more cards so that every ball swishes into a hoop of the same color. The cards may be rotated and/or flipped but must lay on top of each other in the same orientation, and no hoop or ball can be left unmatched.

From your reply, I'm not sure you understood my question (maybe I should take pictures of 4 cards to illustrate). Let me state it in reverse. If you have what seems to be a valid 4 card swish per the rules you've quoted, is it still valid if the 4 cards can be rearranged into 2 sets of 2 swishes?

@JoeGolton, I did slightly misunderstand you. You are asking if 2 sets of two card Swishes that could be scored individually, can be scored all at the same time as a 4 card swish if the hoops and balls don't overlap among all 4 cards. I say the answer is probably yes, but you are correct that the rules don't give an example that makes this perfectly clear. The video doesn't help either http://www.youtube.com/watch?feature=player_embedded&v=Hzi28pbeWWw#t=75s

I read over the rules several times and saw the video as well and ended in the same place you did. Probably yes. However, it does seem odd because it is easier to find 2+2=4 swishes than a 4 swish that can not be decomposed into 2 sets of 2. Meanwhile, the scoring treats these two situations identically.

I think that while the rules basically specify a valid swish has all balls and hoops paired up, that being that there are no incomplete sets, to allow re use of a symbol should be disallowed as it also then implies that you can stack multiples of the same card as part of a "pair"
If you then have 3 or more cards with only 2 completed symbols but "all" are complete you shouldn't get the points allocated for 3 or more, while at least this doesn't break a scoring system that all in all doesn't reward skill, just deviance for circumventing a grind...
I think moving forward an unwritten rule should be that no 2 cards that have overlapping symbols of any colour can be used to form a swish. It satisfies the issue of unfair or devious use of cards, alongside the re-use being unavailable makes the 3+ swishes more impressive and gratifying.
And back on topic, if you have 2 swishes that can play as separate swishes ( but don't overlap on any symbols ) then you have just claimed 2 swishes. There is no indication of a benefit besides getting more cards quicker, while it could be interpreted that players must wait for the dealer to deal the replacement cards before stating swish again, which could mean someone else could get the pair. So in the spirit of fair play, I don't believe that you should be able to claim a 4 card swish made of 2 individual swishes.

And of course actually reading the other answers again and the reply from the creator, I do get why the points is simplified in the original rules, however the rules do actually specify that a single swish may be made at a time, based on the fact that a player calls swish, identifies the swish and collects those cards, and players may not commence until there are 16 cards on the table unless there are no more cards left to be dealt, in which case street rules come into effect and people bring out their knives as they swipe all the swishes they can see at the same time :)

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How to Measure Luck vs Skill in Games?
Anyone who has played a wide variety of games knows that some games are almost purely skill (i.e. Chess, Go), while others are 100 % luck (ie. Candyland, card game of War). However, most games are in between. I would find it helpful as a game buyer if there were an objective measure indicating how much luck is involved in a game, to include as one of several criteria when deciding which kinds of games I'd like to buy and spend time learning. I personally prefer games where luck/probability plays a role, yet deliberate practice increases skill over time (i.e. card game of Bridge). But obviously other people will have different preferences.
I've observed many debates about the amount of luck and skill in certain games and I sometimes use information gleaned form these informal debates to help me decide whether to purchase a game. Very few of these debates cite objective measures to indicate how much luck or skill is involved.
Note that in a BGG luck/skill thread I started, one person did cite an attempt to objectively measure whether skill exists in the game Fluxx.
So what measures exist and how useful are they? Or is there some persuasive mathematics to suggest that useful measures (of how much luck is in a game) are not possible?

I don't think an objective measure for amount of luck exists. Take for example, Texas Hold-em. A single hand of poker is nearly 99% luck. As the size of each persons stack increases, and more hands can be played before running out of chips, the amount if skill increases. But, by what percent? As stacks approach infinity, does skill approach 100%? Now, how do you measure poker luck vs. other games luck? Total randomness?

@user1873 Is there a more rigorous way to state your claim? If so, that would be the basis for a good answer. However, if you didn't click on the link I posted about Fluxx above, I suggest you do so. The person conducting the study took a fairly clever approach to a game which has a very random feel to it: See if a purely randomized opponent would beat an opponent who follows a very small set of simple rules. He then ran a large number of simulations and found that following simple rules wins more often over randomness. I'm pretty sure you'd find the same in Texas Holde-em.

@user1873 That's not a terribly convincing argument - the problem with poker is mostly that there's not a well-defined game length. But for most games, you can simply ask about a single game. For example, perhaps for chess, the best players can win against typical/new players nearly all the time, but in a more luck-based game, they'd win only 75% of the time. Such a number for poker would of course depend on the length of the game, but you could try to find one for a given length. It's difficult to measure, but not impossible to define.

@Jefromi, then you are using for your objective measure, "the best players have better than an equal chance to win against a new player." That is a reasonable enough measure, but how can you be certain that you have a typical new player, and how many games would you have to play to determine if the expert won their 75% of the time due to strategy or random factors?

I read the links, but there is still an issue measuring luck. Balancing for 1st player turn bias is a good start, but what do you measure from as your starting point? (the random player, or a new player with basic strategy) If using the random player, consider a game of 3-in-a-row. On a tic-tac board, a random player should only draw vs. an expert, and lose about 5/6ths of the time (math incomplete). On a 4x4 board or larger, the random player gets worse. The expert who goes first always wins, and nearly always wins going 2nd. Did this trivial change make the game less luck based?

@user1873 Not following what you're trying to point out with the 3 x 3 tic-tac-toe example. This is zero luck game, as the player with perfect skill will never lose. But it is also a low skill game as it doesn't take too much study to master the strategy.

A game of 3 in a row is 100% skill, but how can you determine that. On a 4x4 board, an expert player that goes first should win 100% of the time. Does this mean that the game is 100% skill, or does whatever determines first player introduce luck to the game? My real point was that if you change a 4x4 board to a 5x5 board, a truly random player will get worse, where a basic player won't be affected. If you are using random players as your yardstick, meaningless changes like these will give the appearance of more luck.

@user1873 I did not say that I had proposed a precise definition for a good metric - that's why I didn't post it as an answer. I just attempted to show that a metric that would work for a game of fixed length could also be applied to a game like poker. The fact that it's difficult to propose a perfect metric should not be confused with it being impossible to propose a fairly good one (or a set of them) that would start to put approximate numbers to peoples' sense of luck vs skill. Hard to answer questions are still good questions. (And comments aren't for discussion.)

The fact that this question is already turning into an "extended discussion" in the comments sets off warning bells in my head...

@the: as it probably does for a few users around here (including those who have voted to close). Nevertheless, the discussion is rather off-topic; I would appreciate an answer that outlines a systematic way of describing (and quantifying) the chance and choice of specific game mechanics.

(To clarify, the discussion in comments is off-topic. If someone has a system to share, I'd love to see it.)

I personally think this is a really good question and have been pondering an answer.

Wow it's amazing what these words 'skill' and 'luck' inspire to us gamers :). Anyone reading here searching for an answer should know that.. is the glass half full or half empty?:)

Rather similar: Calculating luck as a factor in board games

If I win, it's skill. If you win, it's luck.

This might be better asked at Stats.SE.

@SevenSidedDie I disagree, to me, the root of this is a fundamental problem that boardgamers face.  IE, did I win because I was lucky or was it my mad skills?

@PatLudwig I don't disagree that it's of significance here, but I do think that people who rigorously study probabilities and human behaviour are going to give better answers to a question about how to rigorously analyse game probabilities. :) This might actually have an established academic answer that stats people could just give off the tops of their heads, for all we know. Game Theory is squarely a stats subject.

Think about the Elo Rating System used to measure the relative skill of Chess players. This is essentially an equation that takes a series of wins and losses and produces a number that can be used to predict the chances of one player winning a game against another player.
One of the inputs to the equation is a "distribution" that describes how much a difference in skill changes your odds of winning. This part is tweakable - they started with a normal distribution, but some groups have switched to a logistic distribution because they've found it to be more accurate.
If we had a fully deterministic game like "Which player weighs more?", the Elo rating would just be your current weight (or some deterministic mapping thereof), and the distribution parameter would be the Heaviside function (0 if less than 0, 1 if greater than 1).
If we modify that game a bit to make a game like "Roll a D6, add that to your weight, and see who has the highest number", the Elo ratings for everyone would still be the same, but the distribution function would have to change so that the behavior from -6 to +6 was a stepwise thing based on the distribution of die rolls.
If we had a zero-skill game like War, everyone would have the same Elo rating (and the distribution function wouldn't matter).
So if we have a two-player game that produces win/loss results, and a set of players, we can force the players to play games against each other and establish Elo ratings for every player, using some distribution function (maybe we start with a normal distribution like chess did). Eventually we'd be able to look at the results of all the games and determine that a logistic distribution (or a log-normal distribution, or some other distribution) did a better job of predicting the results than a normal distribution.
The shape of that distribution - how 'wide' it is, how it tapers near the ends, how steep it is in the center, and so on, is the answer to your question. Of course, this takes a huge amount of effort to measure, but it's theoretically possible to determine that curve.

Addendum: Here are some distribution curves (courtesy of Wolfram Alpha). Apologies that these don't have 100% consistent X/Y axes, but hopefully you get the point.
This is the 'normal' distribution (rather, its CDF). For example, using the yellow line, we'd expect a player with a skill of "+2" to win about 80% of the time.

The logistic distribution is very similar, but has heavier tails (this is really subtle!). The X-axis here is different (why, Alpha?); again using the yellow line we'd expect a player with skill "+50" to win about 90% of the time.

Note that with both the logistic and normal distributions, there is a parameter to tweak how 'wide' the spread is.
Heaviside:

Stepwise (using the "weight + d6" example, this should be transposed up and flatten out at the left/right extremes):

i am not sure iunderstand how different these curves would be. Coul you add diagrams for a fullly deterministic game (like how old are you, what do you weigh), a competely random game, and maybe one or two games that are some where in between?

Nice answer! It's good to point out that even in a "pure skill" game like chess, you still need that distribution. I wonder how things would look if you gave each player a personal distribution--more consistent players would have smaller variance etc. Optimistically, there would be a lower bound on the variance, and that would be the luck factor.

+1 This sounds workable.  Did you definitely mean -12 to +12?  I was expecting something like -6 to +6.

Doh, you're right. I was somehow thinking of a D6 that rolled -6...+6.

I think you're onto something here.  Would it be possible to get some images that are examples of sample distributions outlining how different games might appear?  I think it would help to visualize things.

Note that the parameter for how wide the curve is for e.g. normal distribution, is not really measurable, as you have no fixed scale for the x axis (i.e. what is "+1 skill").
One attempt at an objective measure might be the limit of the distribution as x goes to infinite, i.e. the probability that the most superbly skilled player imaginable beats a "normal player".

@tengfred Elo is based on a rating gap of +200 skill means that one player should win 3/4 of the time (of the non-drawn games), the lower rated player should win only 1/4.

@tttppp, yes, but that means the skill value is defined in terms of the probability to win, meaning that a statement such as "A game is more luck based if the less skilled player has higher chance to win, with the same difference in skill" is meaningless. This is problematic if you are looking for an objective way to measure luck.

@tengfred Yep, I see!  And if we change "same difference in skill" to "best player vs normal player", as you mention above, that does solve the issue.  We could maybe instead look at players who beat each other 3/4 of the time (+200 skill), and measure the length of the longest chain of players (which for a large sample is effectively the width of the spread as mentioned above).

First, there is a quite elusive difference between chance and luck, but it is worth noticing before analysing either concept deeper.  Any non-deterministic event during the game is necessarily a chance component; but its effect on a given game situation may be exactly neutral (either in practical terms in a unique situation, or in some general, mathematically qualified meaning).
For example, the initial setup in Dominion, where kingdom cards are randomly chosen (and made available to all players), is a major chance event that entirely changes the overall course of game; but the luck effects can be almost as small as the mental differences between the players can ever be.  This is because each player gets nearly the same opportunities as everyone else - regardless of what they are.
In games with mixed luck + skill factors, there is a tight interaction between the two.  The skill factor includes knowing the statistical distributions within the chance factor.  There may exist critical but non-obvious breakpoints between distinct strategies, based on analysis or experience, triggered by observable game events.  Conversely, decisions taken by the player can subsequently affect the degree of chance (make the game more random or less random), as well as the expected outcome (bad luck versus good luck).  This does not really mean that more skill could translate to more luck, but it should warn against many naive imperfect metrics of luck versus skill based on what looks like magical luck or enormous chance.
In addition to these obvious "combinatorial modelling" and "statistical modelling" aspects of skill, there are also others.  Strategic (high level models), psychological (basically a statistical model of the opponent's decision making), memory and so on.  Each of these components assigns a difficulty (a learning curve) to every game, and that is a key obstacle to defining skill vs. luck in a player independent way.
To see this, consider Nim (or chess).  Either you are bright enough to see the full winning (or non-losing) strategy or you aren't.  If you are, the game reduces to a pure game of chance, namely the drawing of who goes first - and maybe some amount of psychology if your partner is still somewhere on the learning curve.  If you are not, you are playing a pure game of skill, struggling to be the first player to notice a winning path for yourself.
This reasoning applies to the other components of skill as well.  Some board games are actually employing quite obscure skills, including imagination, dexterity, or deception, and the game experience can be drammatically different based on where on the learning curve, or natural disposition with regard to that respective skill the player group happens to be.
So, you cannot really define a rigorous methodology and end up with a single number usable with all sorts of player profiles, and with all skill levels.  You need to know your "market" as an assessor, and to know your own player group as a consumer when evaluating a game.
However, knowing the players perfectly, games of skill are easy to detect.  You know how.  Have a relative newcomer play against a relative expert (within your expected group).  What's the percentage of games that the expert will win?  This metric however makes slow games appear as comparatively luck based.  There is not enough "tries" to even out the factors that do tend to even out, including chance. So you can improve this metric by playing multiple rounds for a defined typical amount of time and looking at the probability that the expert will emerge with the better score.
Unfortunately, there is no universal definition of an expert.  It means different skills for different games; and different skill levels for different player groups.
(As a side note, many countries attempt to define games of skill and games of chance for legal purposes, that is, to apply different public regulations to these broad groups of games.  In this context, no accurate definition or methodology is ever employed, which is telling.  Only a few games most often played in commercial contexts ever really get classified, and that classification is usually based on cultural perceptions and on business models.  Example again.  If you are really successful in Blackjack due to your memory skill, then the business model (rather than the game structure itself) implies that you are not welcome in commercial establishments where the game is played.  Blackjack is culturally expected to be a game of chance.  You will experience the opposite expectations in a bridge club or in a public bridge tournament, despite that the same card sets are used in both games, and superficially you control the deals themselves even less in case of bridge.)

"knowing players perfectly, games of skill are easy to detect . . ." I was thinking about that very same methodology lately and how it doesn't seem to work. First time I played Dominion, I was one of 2 beginners against two somewhat skilled opponents. I won 1 of my first 4 games. I've seen the same thing more dramatically with my 7 year old son who, after his first 7 games, managed to catch up to his father in skill level, at least as measured by wins per game played since those 1st 7 games. Yet I keep being told over and over that Dominion is a game of great skill . . .

Conversely, many people claim that Stone Age has a great deal of luck. My son has never won a game and I have yet to see someone who has played less than 5 games win against someone who has played more than 30 games. Could all be a statistical fluke of some sort where the noise is drowning the signal. But it does make me wonder about "games of skill are easy to detect" . . .

"games of skill are easy to detect", right? 1) How many games do you to play to know that the newcomer didn't win by luck? How do you know your newcomer isn't especially adept at game x, and is a quick learner? How do you know the experts expected win percentage (or are you comparing their win percentage vs. 50% chance)? This answer appears to be a lot of talk with little actual information. Dominion Kingdom setups can have a huge effect on win percentages, directly affecting the 1st player advantage. So when determining if Dominion is more/luck based, do you have to compare only a ...

particular kingdom setup? Do we need billions of rankings for Dominion because the game rules "change" for each kingdom setup? ... Does this answer actually answer the question, "how much skill is in this game?"

@user1873 - Re statistics: There is no threshold.  The described method also does not work if you cannot guarantee that the player's overall skill levels are different.  The reason is that game outcomes can prove a skill component, if they are statistically too unlikely to permit a "pure chance hypothesis" in a symmetrical game, but they can never prove a chance component.

Yeah, like I said, "not easy to detect."

@user1873 - Re Dominion:  my own very limited experience in a fixed group of 4 players, 3 of whom play only in this group and 1 player has a wider exposition, is entirely sufficient to prove that Dominion has a strong skill component.  Out of maybe 40 games played so far, the "expert" won maybe 35, usually going 3rd or 4th.  This page has a nice graph that tells me that either there is a big advantage in going third or fourth, or Dominion simply cannot be a game of pure chance: http://www.fourmilab.ch/rpkp/experiments/statistics.html

@JoeGolton (and user1873) - yes, I admit, if the game is obviously non-symmetrical, or if you see roughly balanced results, then you get no conclusions from the outcomes.  There is always also the possibility that the particular players are of approximately the same strength regardless of their different backgrounds.  For me, a proof of the skill component is the existence of players who can beat other players in a statistically significant way over a reasonable duration of game play. With chess, I'm reliably defeated in a single game.  With bridge, it does not go that fast.

I do not have a full answer to this question but I have the beginnings of an answer, which I hope is supplanted by a better one.
This answer assumes that game rules are strictly followed, with no cheating or imperfect components. (In other words, no loaded dice, or imperfect dice whose imperfections can be observed after many thousands of rolls, etc.)
The question can be broken down into several simpler questions. The easiest is:

Q1: Is there a way to know for certain that a game has zero skill?

Answer: Yes. Games with zero choice such as Candyland or the card game of War have no choices. If you can't make a choice, it is impossible to apply skill.

Q2: Is there a way to test whether a game with choices has non-zero
  skill?

Answer: Yes, with caveats. As cited in the question, the game Fluxx has numerous random elements but also choices. A study proved skill exists in the game of Fluxx by playing 200 games where one opponent's actions were random, while the other followed a small set of pre-defined tactics. One can imagine a similar methodology applied using computer simulation to any game which is guaranteed to terminate in a reasonable amount of time.
However, all this can do is prove whether skill exists in a game. It cannot prove that skill does not exist, because if a tested set of tactics is poor, then it may appear that the particular set of tactics used does not produce a statistically significant advantage over random behavior. For most games it will be impossible to test every possible set of tactics, so it is therefore impossible to prove using this method that no skill exists in a game that has choices.
There exists a class of games where behaving randomly in some or all situations confers an advantage (i.e. Rock Paper Scissors). So testing whether skill exists should not just be against a purely random opponent, but also an opponent who automatically follows one simple rule (i.e. always chooses Rock in Rock Paper Scissors, or always follows the identical priority pattern when choosing cards in Fluxx). Then the second opponent can be programmed to detect patterns in the first opponent and react the detected patterns. If that confers an advantage, then skill exists.
There is also a trivial subset of games that can be mathematically proven to have no skill: games with choices where the choices are meaningless. For example, imagine a game whose board design is such that the first move is chosen (like which of the 1st 10 spaces to start on for a Candyland-like board), but every choice leads to exactly the same result - a victory (via different paths, depending on the choice). It would be easy to mathematically prove this game has no skill.

Q3: Is there a way to test whether a game has no luck?

Yes, with caveats. If you can program a computer so that it never loses (it gets to choose whether to go first or last to avoid the first/last issue encountered in games like Nim), then the game has no luck. However, the converse is not true. Just because a computer program has not yet been found that will always win a game, does not prove the game has luck - it could be that the program has not yet been discovered.
There exist classes of games where it is clear there is no luck by simple reasoning. One such class is games with no hidden information and no randomization mechanisms, such as Chess or Go. There may be other such classes as well, though I'm not clear as to whether games with hidden information such as Stratego can be considered to have no luck, because there is the luck of which types of information get revealed in which order.
So far, I've covered the very easiest edge cases and not tackled the much more difficult question:

Q4: For any pair of games, both of which (provably) have some luck and some
  skill, is there a testing methodology which will demonstrate which
  requires more skill?

I do not have a complete answer for this much more difficult question. I do not know whether it is even possible to answer, or whether there is academic literature on the subject. But I do have the beginnings of an idea:
How many lines of code does it take to program a computer to never lose? By this metric, Chess requires vastly more skill than Tic Tac Toe - which obviously matches intuition. This has issues such as the relative length of computer programs varying with the computer language chosen, and sequential vs parallel computing architectures (very relevant to pattern matching games like Chess or Go). But if this idea were further refined, it might actually lead to something, even if just a complexity classification system along the lines of Stephen Wolfram's work on complexity in cellular automata.
This is a tough question, and perhaps it won't be answered fully for hundreds of years, if ever. But if there's a better answer than this one, I'd love to see it. And I welcome feedback on how this answer can be improved, if there is anything faulty about the logic.

Regarding Q3: I think absolutely no luck can often be determined based on chance elements and information. Go and Chess have no hidden information, and no random elements, thus they are 100% skill games (regardless of how well computers can play them). Stratego has hidden information but no chance elements. Backgammon has no hidden information, but lots of chance. However, in either of those games, I think a skilled player would defeat a random player  99.999% of the time.

You haven't defined what it means for a player to play with random actions.  For example in poker it is important that a player's bids do not reveal their hand, and so they should bid 'randomly' using a carefully calculated distribution.  Just because the best strategy is to play 'randomly' does not mean there is no skill in the game.

@tttppp You're absolutely right. An even simpler example is Rochambeau (Rock Paper Scissors). I'll figure out how to incorporate into my answer, though it may take some thinking.

@shujaa I agree with you and need to figure out a way to incorporate your comment into my answer.

I think you're asking the wrong question.
Instead of asking about how some nebulous term (such as 'luck') applies to any particular game, what if we ask instead "how much effect can random chance have on the outcome of a game?"  Now we suddenly have an answerable question!
A game like Chess or Go has no random factors in it, so the answer would be "none".  This would, in your terminology, be an "all skill" game.
A game like Candyland or War is completely driven by random factors, so the answer would be "all".  In your terminology again, an "all luck" game.
Obviously the in-between games (such as the oft-mentioned Fluxx) have some measure of both.  The shuffled deck of cards, the roll of the dice, whatever is built into the game to incorporate the random factor.  There are player decisions that are critical to determining the outcome, but the decisions to be made will have to involve the random factors.  Sometimes your numbers come up in Catan, and sometimes they don't.  What you do in each case, how you change your play to extract the most resources out of that random factor, that's a good portion of the skill involved.  How can we measure how important the random factor is in any of these games?
Well, I think our best method of doing that is by applying a very powerful tool: gamers.  It's the near-fanatical players of the game that have the knowledge and experience required to 'rate' any particular game that they've played sufficiently.  In exactly the same way that it's the gamer that rates a game on the 1-10 scale, it's the gamer that would be able to rate a game on much much of a factor random chance has on the outcome of the game (I would envisage this as a 0%-100% scale) and via a dozen votes I'm sure a common average value would again emerge.
In summation, my answer on "How to measure luck VS skill in a game?" is this:
By adding a "Impact of randomness on game outcome" value to every game in the BGG database and waiting for the most powerful tool of all (us) to calculate the answer for each and every game.
And I really don't imagine that there's any other method that can work anywhere near as well.  It's the power of parallel processing!

Though BGG does not have a ratings category for either luck (randomness) or skill, I've seen many text descriptions. I am led to believe from the text descriptions that gamers, being human, are a poor judge of randomness. Classic experiment: ask 10 people to distribute themselves randomly throughout a large room. They spread themselves evenly apart, no clustering. Predictable. And very non-random. To site specific game example:

Stone Age is a game where numerous people who have played less than 20 games comment about how much luck it has. Yet highly skilled Stone Age players never lose to people who have played fewer than 20 times. So if you have 1000 people rating the game, only 50 of whom have achieved a high level of skill, their voices will be drowned out by the other 950 low skill players that aren't good enough to realize that the role luck plays in this game is small once you reach a certain skill level. The dice rolls and random cards/tiles ordering sure gives it the appearance of luck, though.

I know exactly what you mean and I've wanted some sort of ranking/rating system as well, in part because how "mean" a game is depends in part on how much luck is involved (mean games being ones where non-winning players feel as though they lost due to choices targeted at them by other players; Diplomacy is probably just about the meanest game out there).
While a very granular or statistical study comparing and ranking games is probably more work than it is worth, I think that there is a rating scheme that would be useful for knowing more about a game before buying it. How about giving a rating to the existence of various elements in the game based on whether they are

non-existent or flavor-only (-)
balanced (according to the rule-book) variety-increasing elements (1)
asymmetrical and capable of altering scoring position (2)
the core mechanic for winning (3).

The elements would be

A: Unchosen Flat Chance (single die, spinner with equal segments, etc.)
B: Unknown Probabilistic Chance (dice under a cup, cards dealt face-down, etc.)
C: Hidden or Simultaneous Choice (pre-selected cards, simultaneous signals, etc.)
D: Known Probabilistic Chance (pair of dice, deck of known cards, etc.)
E: Choice with Asymmetric Knowledge (pieces with values known to less than all parties, etc.)
F: Fully-informed Choice (chess capture, etc.)

So, Chess and Go would be A-B-C-D-E-F3. Rock-Paper-Scissors would be A-B-C3D-E-F-. Stratego would be A-B-C2D-E3F2. Texas Hold'em would be A-B2C-D3E2F2. Candyland would be A3B-C-D-E-F-. Diplomacy would be A1B-C3D-E-F-.
This is a first draft, and I'm sure there's room for improvement, but I feel like you could look at the distribution of numbers by element to get a sense of the game. That said, my ordering or categorization seems lacking when it comes to Diplomacy.

An objective measure for Luck doesn't exist, so no such comparison can be made.
What is luck, in game terms? One definition is, the chance that you can win or lose a game, due to random chance and not due to decisions that you or your opponent made. A game of 100% pure luck, cannot have any decisions that you or your opponent make have any impact on who wins (i.e. You should be just as likely to win/lose by making random moves). A game of 100% pure skill, cannot have the result of who wins or loses determined by anything other than decisions that you or your opponent make, (i.e. Your decisions should cause you to win/lose more often than random moves).
Luck can be equated with randomness. An objective measure of luck would be:

The difference between how often you win a game compared to how often a random player would win a game.

While this is a completely objective, it isn't very useful. Some examples are in order:
A very simple game
This game consists of a shuffled deck of 3 cards labeled 1, 2, and 3. The game is played over two rounds, with the deck shuffled between rounds. Each player chooses a number between 1-3, then the top card of the deck is revealed.

If the card is >= to the highest guesses: the player with the highest guess scores the other players guess (if both are the highest, both score)
If the card is < one player's guess: the other player scores the high guess.
If the card is < both player's guesses: neither player scores.

Random Play: There are 27 possible outcomes in this game each round.
          ¦(1,1)¦(1,2)¦(1,3)¦(2,1)¦(2,2)¦(2,3)¦(3,1)¦(3,2)¦(3,3)¦Guess (Player1,Player2)
          +-----+-----+-----+-----+-----+-----+-----+-----+-----+
      R(1)¦ 1,1 ¦ 2,0 ¦ 3,0 ¦ 0,2 ¦ 0,0 ¦ 0,0 ¦ 0,3 ¦ 0,0 ¦ 0,0
      R(2)¦ 1,1 ¦ 0,1 ¦ 3,0 ¦ 1,0 ¦ 2,2 ¦ 3,0 ¦ 0,3 ¦ 0,3 ¦ 0,0
      R(3)¦ 1,1 ¦ 0,1 ¦ 0,1 ¦ 1,0 ¦ 2,2 ¦ 0,2 ¦ 1,0 ¦ 2,0 ¦ 3,3

An expert player will never choose 3 for the first round. Doing so would result in their opponent scoring 3 points 33.3% of the time (ensuring a loss), drawing 4/9ths of the time, and only gaining points in 2/9 of the outcomes. An expert will never choose 3. If we compare a random player to an expert:

Rnd1: E1,R1 = 3/9 Win, 2/9 Lose, 4/9 Draw
Rnd1: E1,R2 = 5/9 Win, 4/9 Lose
Rnd1: E1,R3 = 5/9 Win, 4/9 Lose
Rnd1: E2,R1 = 14/27 Win +  6/27 Lose, 7/27 Draw
Rnd1: E2,R2 = 3/9 Win, 2/9 Lose, 4/9 Draw
Rnd1: E2,R3 = 14/27 Win, 8/27 Lose, 5/27 Draw

The expert player will open with Choose 2, and will win nearly twice as often as the random player. So, we will give this game a score of 60% skill (didn't work out the exact math for how often a random player draws, something close to 16%). So let us change the game. Instead of 3 cards, we will have 10 cards from 1-10. Now the random player has more bad choices. Instead of just 1/3rd of the choices being bad, now 4/10ths are bad. Is the game anymore skill based? An expert player will improve their win percentage against a random player, but what does that really mean?
We can make other rather trivial changes to the game like, not shuffling between rounds or removing several cards face-up from the shuffled deck (10-card version) that would increase an expert players win percentage versus a random player. These objective measures just don't have any meaningful value when comparing these trivial guessing games. If you take the fact that when trying to objectively measure the luck in these simple games, you can see the trouble of trying to apply these measurements to games with wildly variant forms of randomization.

You've convincingly demonstrated how one particular objective measure is barely (or perhaps not at all) useful. However, this does not prove that all possible metrics are useless. One I've been pondering since I read your answer: how many lines of code does it take to program a computer to never lose? By this metric, Chess requires vastly more skill than the game you describe - which obviously matches our intuitions. But the 10 card version of your game vs. the 3 card likely requires the same amount of code or only slightly more, suggesting that the skill required has not increased.

@JoeGolton, It is impossible to prove a negative. Chess, and 3-in-a-row are both games of 100% skill, but trying to use this particular measure to determine how much more skill exists in that game than this game is useless. The only measure that is possible is a 3 number scale, all luck, all skill, or somewhere in-between. As to the 10-card/3-card game, I was only illustrating that trivial differences can make large changes in wins vs. random players (it still isn't clear what the optimum first move is to me 5, 6, or something else). I probably need more and better examples to show the jump.

Let's say it takes 500,000 lines of code to program a perfect chess player, vs. 20 lines of code to to program a perfect tic tac toe player. Why is this difference a useless measure? It may not be a truly great measure, because it probably doesn't say much if a game takes 20 vs 30 lines of code. But for wide swings of skill - doesn't several orders of magnitude more code strongly suggest more skill required?

For example, the win rate was about 30% for a random player in this 3-card game where Fluxx had a win percentage of 14%. So is this game more skill based then Fluxx? What if we add more players to Fluxx, or compare Fluxx to the 10-card game. As for lines of code, that is a terrible measure of complexity. Complexity itself might be a good measure of skill for a game, but now you have another problem. How do you measure complexity?

I agree with @user that lines of code is a terrible measure of complexity, but I think it's an even worse measure of luck.

Consider the game of Risk.  In a 1v1 game of Risk an expert player can beat random play very close to 100% of the time.  If this were a measure of skill than risk would be 100% skill. But risk, obviously, isn't 100% skill.
Breaking down this question should be done differently.  The question is how likely is it for a player who plays worse to beat a player who plays better(games with more than two results I'm ignoring for simplicity).  This is conceptual and their is a lot of nuance in what exactly you are measuring but I think it really is what you are actually trying to get at.


text:
Do the vast majority of Rat-a-tat Cat rounds end very quickly?
My son was gifted this game today. Given that we play a variety of complex games in our family (including Bridge), we're very surprised how much we like it and the various skills it brings into play (probabilistic assessment, poker face, memory, etc.). We're also surprised at how rounds usually end so quickly, as the rules provide for getting to the end of the deck. We have yet to go half way through a deck.
Is this typical, or are we missing something to the strategy?
Our probabilistic reasoning:
A deck consists of 4 each of 0 through 8, 9 9's, and 3 each of the power cards. So an average dealt hand has 20 points, with 10 points visible (I'm counting the power cards as 5's given that you'll have to draw randomly from the deck at the end for each of these).
A typical starting hand might have a low number and a high number known upon deal. If your first draw is a low number, you replace your high numbered card with that one and then you call for the end of a round. Why? Because you have, say, 5 points on your ends and an assumed 10 points (it's random) in the middle, for perhaps 15 total.
You have no knowledge of other hands but on average each will be 20, and with one more draw will be reduced by 4 points to 16 (on average). Not a huge edge, but definitely an edge. And this only widens as the sum of your two end cards drops further below 5.
Am I missing something? Or is it correct to call for the end of the round if the sum of your two end cards is 5 or fewer points after you draw the first card of the round?
The reasoning gets a little more complicated for subsequent rounds as other players' hands improve, but similarly if you know that 2 of your cards are very low, or better yet 3 within the first few rounds, then you call for an end, right?

Yes, you should call for and end or knock-out as it is commonly called.
Your reasoning is sound. With 54 cards in the deck, 9 of which are special non-point cards, and four cards each of 0-8 plus nine 9s, the average hand is 20 points. There in no way of knowing what your opponents hands are worth, except in the rare case that you draw 1 of 3 peek cards and then you draw 1 of 3 swap cards. (In the children's variant, all players two outermost cards cards are revealed).
Since you cannot know what your opponents have, it is best to call for the end of the round when you know that the sum of 2 of your cards is less than 7, or 3 of your cards are less than 12. (Edit: My initial calculations didn't take into account that your opponents get to look at one more card, and reduce there score as well.)
The only thing that might change this strategy slightly (and I am not certain if it does change the optimal strategy), is as the number of players increases the chances that one or more of your opponents draws a Swap card during the knockout round also increases. If your opponents know that you are using this strategy of only knocking out when you have < 7, or < 12 when you replace a hidden card with a drawn card, they will be sure to Swap cards with you (most likely they will swap with the card that you drew to replace your other card). They will know that they have a better chance of getting a '4' or less from you, then from drawing from the deck.
As for conclusion that each player would take less than six turns on average, I am not certain. I haven't worked out the math for it, but the number of turns on average would differ depending upon the number of players. If there are more players playing with the < 10 points sum total of known cards, then the chances that at least one of those players has a low hand increases.
This analysis was only performed for the first turn. I suppose you could calculate the AVG number of points that you would reduce your score per turn, and use that to figure out when to go out on later rounds. For the first round, with only looking at 3 cards, you will reduce your points on average by 2.63 points. This result was arrived at by looking at a million random 3 cards from the deck (with wilds removed, so the odds might be slightly different), and choosing the lowest 2 cards to keep for a single player. I did not examine what happens as the number of players increase, but my guess would be that as the number of players increase, the chances that at least one other player is able to reduce their score by more than the average increases (i.e. with two players, it is probable that one player reduces their score by more than 2.63 and the other by less than 2.63. For three players or more, the odds that someone is able to drop a '9' or other high number for a '3' or other low number increases, and that lucky person will have on average less than your total.

Your answer cites <10 while I had calculated <6 (based on the idea that a typical last draw by opponents will drop their score of 20 by 4 (i.e. a 9 gets replaced by 5). Can you elaborate as to why 10 as opposed to 6?

@JoeGolton, I did make a mistake in my calculation. If we only look at the deck without the wild cards, then I think on average your opponents will be able to reduce their score by 2.63 points. I might work on the more correct answer (using wilds), but not soon. The Peek card is worse than useless during a KO round (you don't even get to swap a card with the deck), the swap card is probably easy to handle programically always stealing a card you swapped with the deck, but the Draw 2 card is more complicated (you have to calculate the odds that giving up your first draw might improve with 2nd.

Thanks for correcting. Interesting that my guess of "4" was so far off from your 2.63 derived from simulation. BTW, by my interpretation of the rules, you can swap the peek card into your hand if you want. So a "peek" card is therefore not "worse than useless." You can replace a 9 with it, which in turn leads to a random draw, whose average will be around 5. Not sure if that means the 2.63 increases slightly.

@JoeGolton, I meant that during the KO round drawing a peek card is useless, the rules say "When you draw a Peek card, show it and then peek at any one of your cards." You clearly cannot choose to replace one of your outermost known '9' cards with a drawn Peek card. An alert opponent with good memory would remember where you placed that drawn card and accuse you of cheating.

I just carefully reread the rules and you are correct - I was doing it wrong. For such a simple game, it is remarkable how poorly the rules are written. I've now read through them at least 4 times and every time I've discovered things I was doing incorrectly.