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Analysing the Small World Underground Contest - Non-Uniform Distributions

Geoff Thomas
United Kingdom
Enfield
London
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In my previous blog post I attempted to analyse the current Small World Underground Contest. For a first attempt I used a uniform random distribution for assigning votes and as expected the first comment disputed that this was realistic. At the time I refuted that I didn't think it would make much of a difference to the outcome but I thought I would run some further simulations to see if this held true.

Firstly I thought I would look at some rather shallow distributions (red and green) and compare against the uniform distribution (blue). I used cut-offs of 2500, 3000, 3500, 4000, 4400, 4800, 5000, 5200, 5400, 5600 and 5800 votes.



This horrid mash of colours shows that there is very little difference between the simulated results. I collated some data on the mean number of winners based on the number of races that were remaining and produced this graph which shows the number of winners against the number of races remaining when they entered.
.


Once again, all very similar although nowhere near what I said in my prediction blush. Seeing as this didn't provide much of a difference, I thought I would widen the probability distribution and tried the following: (new distribution in red, uniform in blue).



Now this looks considerably different and provides a very different winners graph.



This clearly shows that these weighted probabilities cause many more winners to occur while all 15 races are still active. Obviously in this case the popular races are voted on much more often early and many of the 10 winners are already decided by the time the competition nears finishing with 50% of the winners already allocated by the time the first race is eliminated. Interestingly, the advantage to voting just after someone is eliminated as opposed to just before someone is eliminated is much reduced as well and overall the graph is pretty flat. However, people who vote once a few races have been eliminated have a similar chance to those who voted when all remain, they just produce fewer winners since fewer people vote in this time period.
Since it was obvious that the probability distribution had a significant effect on the winners graph, I thought I'd run through some more simulations with other distributions.





All in all, quite similar results and an even starker contrast to the medium spread distribution. They also all generate very similar winner graphs whereby nearly all (and in the sharpest case, all) the winners are decided before the first race is even eliminated. However, what this graph doesn't show is that with the steepest of distributions, these winners were all decided within the first 1000 votes.



So it seems I was wrong about non-uniform distributions, it seems that they can heavily influence when the winners will be decided. Unfortunately I don't know what the actual distribution is with the current votes but from the data gathered it seems that anything more than a very shallow probability distribution will mean most of the winners will come from those who have already voted. Once races begin to be eliminated, we will have a much better understanding of the distribution as we will be able to compare how many votes the excluded race has against how many votes have been cast.

However, this isn't the end of the story and those who have yet to vote certainly aren't out of it yet. I have begun looking into mixed strategies whereby people entering later purposefully vote against the expected probability distribution in order to improve their chances of winning and I hope to be able to present some results from this tomorrow.
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Tue May 10, 2011 8:22 pm
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Analysing the Small World Underground Contest

Geoff Thomas
United Kingdom
Enfield
London
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I often partake in the contests that appear on the geek. Even though most of the time the odds of winning are actually quite low, since it's free to enter there's not much to lose.

The latest contest on offer for the Days of Wonder's Small World Underground piqued my interest, not for the prize, but the way in which the contest is being carried out. I wasn't around in 2009 when apparently a similarly previous contest was run but this certainly seems like an innovative way to run a contest. But unlike most other contests that require you to answer questions, this contest relies on you picking the correct order of a popular vote. This is almost a game within itself and as such, I thought I would analyse the contest to see if there was any strategy to entering.

Before expanding on my analysis, I feel that I should add a disclaimer that I have already entered this contest.

Firstly, lets look at the facts of the contest:

- The contest is asking you to pick the correct order of 3 races from a set of 15. Hence, there are 2730 distinct combinations of answers. (nPr = n! / (n-r)! which in this case is 15!/12! or 15 x 14 x 13).
- When someone votes, their first choice is given 3 votes, their second choice given 2 votes and their 3 choice given 1 vote. Therefore each person casts a total of 6 votes.
- During the first 3 days of the contest, all 15 options will be available. After this period, and each day after this, the lowest ranked option will be eliminated from the contest until only 3 remain. At this point, the votes will be revealed and winners announced.

Secondly, I have made the following assumptions:
- Votes will be carried out randomly. Nobody will have any knowledge as to the current totals and people will not be swayed by earlier voters pushing their own agendas.
- I have assumed that 6000 people will take part in the contest based on data from previous contests and the current number of the contest microbadge owners. Looking at the results however, the number of votes seem somewhat irrelevant as to when is the best time to vote.
- I have assumed the distribution of votes will cause entries to be eliminated after 2500, 3000, 3500, 4000, 4400, 4800, 5000, 5200, 5400, 5600 and 5800 votes.

Using these assumptions I have carried out a series of simulated contests (100000) and compiled the results in the graph below.



The graph represents the number of times a voter won according to their position in when they submitted a vote. I have contracted the x-axis to not show the first 2000 votes as it's pretty much a straight line. From this we can see that if you vote towards the end of the contest, you have a much better chance of winning than if you voted at the start. As such, I made some adjustments to the cut-offs based on the fact that if people know this knowledge, they would delay voting until much later to improve their chance of winning.

So I changed the voting cutoffs, this time to 2500, 3000, 3500, 3700, 3900, 4000, 4150, 4300, 4500, 5000 and 5500. This gave a graph as follows:



I took this to an even greater extreme and changed the voting cutoffs to 2500, 300, 3200, 3300, 3400, 3500, 3600, 3700, 3800, 3900 and 4000 which produced



This last graph is actually drawn to a different axis (5000 as opposed to 2000). Overall though, it look pretty grim for those of you like myself who have already voted (In hindsight I should have carried this analysis out before entering...) and we can but only hope that we are lucky enough to find ourselves with the winning combination (a 1 in 2730 chance). For those of you who haven't voted, the evidence certainly points towards waiting it out until only 5 or 6 competitors remain and getting your vote in as soon as possible after a knockout has occured as this gives you a much better chance of winning. Interestingly, 6000 participants over the course of the competition is about the threshold required to get 10 winners.

I hope to do some follow up analysis as the competition goes on to get more accurate numbers for when dropouts are occuring. However, I can only see more participants causing any radical changes to the strategy that needs to be employed (leaving it too long will mean each combination already has 10 winners thus people will have to enter earlier to maintain any chance of winning). It would be interesting to add an element of preference so that some of the choices are considered more favourable and if this has any affect on strategy but I doubt it would be significant.

It certainly would be interesting to see how people voted when the live results were available. Under these conditions I imagine elements of game theory would begin to become significant in choosing a strategy to win (assuming people are actively trying to win/improve their chances of winning) but with the blind results I think the only variable is the time that you enter.

As a closing thought, I shall make my prediction now that the winners will be distributed as such:

1 winner who voted with 4 races remaining.
3 winners who voted with 5 races remaining.
3 winners who voted with 6 races remaining.
2 winners who voted with either 7 or 8 races remaining.
1 winner who voted with 9 or more races remaining.
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Mon May 9, 2011 5:05 pm
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Ridin' Solo

Geoff Thomas
United Kingdom
Enfield
London
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This week I've been away from home on a course so I've been stuck in a hotel room for most evenings without my creature comforts. I did however take with me my trusted iPad and Le Havre. I've never had a laptop or any sort of portable media device before so I really enjoy being able to take away some tv/movies and such to watch while I'm away. I also have a few board game apps on my iPad, namely Small World 2, Hive and New World Colony.

But as I sat in my hotel room I couldn't help but think that a lot of board game apps are going about things the wrong way. I subscribe to the excellent iOS Board Games blog and from what I read a lot of iOS board game implementations are based on the principal of allowing us to play with other people around the world but this really isn't what I'm looking for from a iOS implementation of my favourite board games.

As I mentioned, I took Le Havre with me while I was away as it's one of my favourite games to play alone. Le Havre was my introduction to "Euro" games and I enjoy it greatly and while it plays great multiplayer, I can't help but enjoy it more when played solo. Played alone the game offers a great "puzzle" experience, essentially asking you to "solve" the puzzle to create the best score. I also think playing alone is a great way to improve your overall ability at the game and I've noticed a considerable improvement when playing against others after solo play.

I often take Le Havre away with me when I go away but rarely do I actually sit down to play Le Havre (and this time was no exception). This is normally because I rarely get a large enough chunk of time to sit down and play the game in one sitting (I don't really like to leave pieces out) and the long time it takes to set up and pack away the game. Both of these problems though could be easily solved by a computerised implementation of the game but most lack this solo play option, instead allowing you to connect with other players or "passing" the iPad amongst you.

Now I appreciate that Le Havre is a special case whereby it can be played solo but I really wish developers would invest more time and effort in producing an effective AI so I can play the game on my own. If I want to play a multiplayer game of Le Havre then I would much rather arrange a time to sit down with my group and chat while we played a few games. While I enjoy playing games, my enjoyment comes from the social interaction of playing, not just playing the game. What I'd like to be able to do however it take my game collection with me while I go away and be able to play them without having to wait for other people to take their turn, worrying about my wireless reception dropping out, having to leave because I'm going out for a drink and play a turn or two before heading down for breakfast.

Now all the board games I own on my iPad do this to some extent (although I wish Small World supported more players), but I just hope other developers will take note and do so as well rather than continue to trend of promoting the multiplayer experience.
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Fri Mar 25, 2011 6:40 pm
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The Gaming Group Veto

Geoff Thomas
United Kingdom
Enfield
London
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My gaming time has been at a premium recently so when I do get a chance to sit down and play some games, I try to play games that I really like. Unfortunately as a gaming group we have eclectic tastes and while we mostly enjoy the same games, some stuff just doesn't go over at all well with some of the group. For example, I really enjoy Cosmic Encounter but one of my friends really doesn't like it. Consequently, I don't often suggest it since I know he doesn't like it.

Now there are a few games that I don't really like to play and most of them are shared by my group, but there are some that my friends really like and I don't. Age of Empires III: The Age of Discovery is one of those games. Unfortunately, as a gaming group we're quite an undecisive bunch and since one of my friends suggested a preference towards playing Age of Empires I reluctantly agreed to play. Now Age of Empires does a lot of things right and would be a great worker placement game except for two fatal flaws. The discovery mechanic and the initial turn order.

Now I'm not trying to use these as an excuse for my poor gameplay since I didn't lose because of them, but I feel my friend did. He was last on the first turn whereby almost everyone's first action is to take a building. Now in subsequent turns you can move yourself up the turn order to get first pick, but randomly choosing the turn order to begin the game just seems like a unfair advantage for getting the best buildings. Why they couldn't have implemented some sort of bidding system to allow you to bid to go first is beyond me.

Secondly is the horrid discovery mechanic. For a game built around completely open information, Age of Empires decided to implement a huge luck based element. So yes, while I can appreciate that there is an element of risk/reward, it just doesn't fit into the rest of the game whereby the only risk you take is not grabbing a set action first. We already play with an adjusted rule whereby you only lose half your points worth of troops it still feels like Lady Luck is playing too much of a part. My friend must have missed 4 discoveries by only 1 person each time and lost 10 or so VP because of it which may have won him. Not only that, some of the discoveries are simply better than others while equally difficult (I appreciate there is a difference between the cards and chits, but each subset has ones which are better than others in it). If you are going to have random elements, at least balance out the gain from the similarly difficult elements so one discovery isn't clearly better than another.

The more I play Age of Empires, the more I dislike the game. I feel bad "vetoing" it and refusing to play because I just feel like I'm a small child throwing my toys out of the pram but equally I really don't enjoy playing it and would rather be doing something else. But it would be hypocritical to veto a game I don't want to play and then be upset when others veto games they don't like. I don't know what the answer is, all I know is that the black ships in Cosmic Encounter: Cosmic Conflict look really cool and I want to play with them...
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Sat Mar 19, 2011 9:07 pm
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"You are a Cylon"

Geoff Thomas
United Kingdom
Enfield
London
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4 simple words that fill me with dread whenever I pick up my loyalty cards and read them. It's not that I don't enjoy Battlestar Galactica, I really enjoy the game, I just don't enjoy playing a Cylon. My friend recently picked up Battlestar Galactica: Exodus Expansion to complete the BSG set and while it definately adds to the base game, it still doesn't make playing Cylon any more fun for me.

Perhaps I'm just not a very good Cylon and as such I never feel like I'm doing anything. When I'm a human player, I'm sitting there trying to figure out who the Cylons are, who is acting suspiciously and who I should be trying to put in the Brig. When I'm a Cylon, there just isn't this aspect of the game since I am a Cylon and I've never found it easy to let my Cylon counterpart know I'm a Cylon without also revealing to the human players that I am.

Secondly, I've never really felt it that easy to throw negative cards into skill checks without giving myself away too easily. As such, rather than be that active in skill checks, I instead take a rather passive approach, generally conserving good cards I have and getting rid of them at inappropriate moments or flooding easy skill check with lots of points.

Lastly, after revealing I am a Cylon I find my input even further reduced. Drawing so few skill cards limits the ability to add cards to skill checks; Most of the time is spent watching the human players Executive Order each other so they get two actions each; My actions are generally limited to drawing or playing a Super Crisis Card (although the new Cylon Fleet board has added some interesting additional options).

Perhaps I'm overly cautious in my non-revealed state because I fear so much the mundane play of being a revealed Cylon. As such, I don't want to "ruin" my game by being revealed early in the game and play a rather passive game doing very little.

I don't feel that the game sits on the tipping point of the scales, possibly tipping one way or the other. It's either overly easy with some indifferent crisis cards or we're in trouble right off the bat and always trying to recover. Rarely do we scrape through on skill checks or just fall short, we either decide to abandon all hope of passing or pass with flying colours. I just don't think the game has small shifts, allowing the Cylon to subtly take the game away from the humans. It has the extremes of easy or hard and you have to use an overt action to hopefully swing it from one extreme to the other.

But it'll still be hitting the table again because for some reason, I just can't stop hoping someone else will be a Cylon and I can oust them and save the day. So say we all.
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Tue Mar 15, 2011 12:19 pm
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Lord of the Rings: The Card Game

Geoff Thomas
United Kingdom
Enfield
London
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The Lord of the Rings: The Card Game has been on my wishlist for a while now. I was a long time player of Decipher's Lord of the Rings TCG which sadly died out and I've been looking at some of the other LCG games that FFG have been producing but I've yet to take the plunge on any. So far however, most have been competitive affairs and while I'm lucky to have a gaming group, I rarely get to play any 2 player games so felt these other LCGs wouldn't see much play time.

The Lord of the Rings: The Card Game seems to be bucking this trend though with both it's solo and co-operative elements which has made me much more inclined towards purchasing. I'm far more comfortable investing in it knowing that I can play solo and that it will be easier to get to the table if 3 or 4 of us can play at the same time.

There's been little information so far on the game but today I came across this and the first thing that sprang to mind is just how incredibly difficult this game sounds. Now I appreciate there's a lot more information to come out and I imagine there are a lot of cards in the player's deck to help them out but it still looks like it's going to be really challenging to beat.

Each player gets 3 heroes and for every player they'll turn over one card of the encounter deck. You'll need to exhaust one character to at least have a chance of progressing so you're left with 2 others. If it's a monster, your then probably going to have to fight it. You'll need to exhaust one character just to defend and then you've got one left to attack back with, probably causing one or two points of damage. Now I know you will be able to play extra cards but from the resource cost it looks like these will be max two a turn (and you only draw one card per turn by default) so it seems things are soon going to be stacking up pretty quickly to keep progressing and kill any monsters that may be coming your way.

There's mention that if you lose a hero there are ways to get them back, and FFG seem to suggest you can still win even if you lose heroes but I'm struggling to see how you can ward off an ever increasing number of things with fewer heroes. I'm also not sure I'm keen on higher threat monsters not being engageable until a player meets that threat. I'm not sure I want monsters hanging around in the staging area as you'll need to devote more and more characters to be able to progress leaving you with fewer characters to deal with them when they enter play. I also feel it will require a lot of threat management to prevent one person in your team being out in front in threat terms. Whoever is "leading" is likely to get all the staged monsters compounding their high threat position by overwhelming them so it'll be interesting to see what mechanics there are to help out other players.

Overall though, the information coming out is building up my anctipation for this game. I hope there will be some replayability but by the sounds of it the new quest decks will add some variation to the game. Here's hoping it can live up to my expectations.
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Fri Feb 18, 2011 8:18 pm
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Plants vs Zombies

Geoff Thomas
United Kingdom
Enfield
London
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So a friend of mine linked me to this article yesterday about the upcoming N/A based upon the popular video game of the same name. There doesn't seem to be much information out about the game currently but I'm a big fan of the video game so this immediately hit my wishlist, especially if it meets it's price point of $20.

I'm a little disappointed though that all the news coming out doesn't seem to suggest there will be any rules for a solo play although one may ask the question, why not just play the video game if you're going to play single player. Still, it looks like a fun adaptation so it's definately one I'll be keeping an eye on in the coming months.

There's some more info here courtesy of
Anders Pedersen
Denmark
Copenhagen N.
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Thu Feb 17, 2011 5:53 pm
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Can Dominant Species dominate our group?

Geoff Thomas
United Kingdom
Enfield
London
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As this is my first blog post, I'd thought I'd start off with an introductory blurb for anyone passing by who may be reading this. Firstly my name is Geoff, I'm 26 and I live in Bristol, UK. I've been a long time CCG player (Star Trek/Lord of the Rings) but moved into "mainstream" boardgaming about 3 years ago. We have a small gaming group that meets once a week but now also game more often with friends during the weekends. We play a wide variety of stuff but my interests lie mainly towards efficiency games and the medium weight euros but my modest collection is quite ecletic. I'm also a big fan of Werewolf. I've never actually played the game "in the flesh", only moderated which I personally find great fun but I've played a fair few games over IRC using my own IRC bot.

So that's me out of the way, now onto the boardgaming. This weekend, I was excited to be able to get Dominant Species back to the table. I picked this up a couple of months ago along with Poseidon and neither of them went that well first time around. We had originally tried Dominant Species with 6 but after learning the rules and a couple of hours of play we abandoned the game as people were getting tired. Since then it hasn't seen the light of day with me until yesterday. I had however leant it out to one of my gaming friends who had managed to get a few plays out of it with a different group and told me it was an excellent game.

We had a 4 player game (myself, Sam, Mickey and John) and I felt good about my prospects after being informed that my opponents had been up until 6am playing 1860: Railways on the Isle of Wight but unfortunately it wasn't to be. The first time I played I had Arachnids and after a few turns it clicked as to what a good Arachnids strategy would be. Get on tundra and score the survival bonus points from these and this is what Sam did for most of the game. This time however, I drew Mammals and for some reason there didn't seem to be any good strategy popping out at me. There were a few meat tokens coming up in Abdunance so I was focusing on getting these on the board in hopes to dominant some regions and I tried to spread myself out to score well in other people's dominations since I would win any tie.

Mickey had drawn amphibians and initially looked in a very poor position. A catastrophe and ice sheet on the first turn had left him with only 5 species on the board but he soon recovered, wanderlusting several good tiles with water elements and dominating them several times. I couldn't tell you what John's strategy was, I'm not even sure John could tell you what John's strategy was other than to completely copy my element choice. John had drawn reptiles but got stuck at the bottom of the turn order quite soon into the game and lost a lot of sun elements during wasteland early in the game.

Ultimately though Mickey came through to win by quite some margin, probably around 30 or so points. I think it would have been a lot closer but "Cold Snap" on the last turn caused Sam to lose his survival bonus which would of got him 28 points and he was dominating many regions for "Ice Age". This did however let me sneak into second place scoring 28 points on "Ice Age".

So will Dominant Species be a hit with our group? I'm not sure. Don't get me wrong, I like the game and I want it to be a hit but for me it just lacks that something to make me really want to play it. I think some of that is down to the fact that I don't really "get it" yet and at the moment I don't really want to muddle my way through the 4 hour play time not really knowing what to do, but I just can't put my finger on it as to why I can't love this game. I want to love it, I want to consider it great, but I just haven't fallen for it yet. Hopefully I can get it to the table again but there's just so many other games I'd rather play again first that I'm not sure when it will see the light of day.
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Mon Feb 14, 2011 11:08 am
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