## Weak Superhero: how to win and lose at Marvel Top Trumps

Top Trumps is a card game for children. The mind can wander when playing such games with kids… typically, I start thinking: what is the best strategy for this game? But also, as the game drags on: what is the quickest way to lose?

Since Top Trumps is based on numerical values with simple outcomes, it seemed straightforward to analyse the cards and to simulate different scenarios to look at these questions.

Many Top Trumps variants exist, but the pack I’ll focus on is Marvel Universe “Who’s Your Hero?” made by Winning Moves (cat. No.: 3399736). Note though that the approach can probably be adapted to handle any other Top Trumps set.

There are 30 cards featuring Marvel characters. Each card has six categories:

1. Strength
2. Skill
3. Size
4. Wisecracks
5. Mystique
6. Top Trumps Rating.

What is the best card and which one is the worst?

In order to determine this I pulled in all the data and compared each value to every other card’s value, and repeated this per category (code is here, the data are here). The scaling is different between category, but that’s OK, because the game only uses within field comparisons. This technique allowed me to add up how many cards have a lower value for a certain field for a given card, i.e. how many cards would that card beat. These victories could then be summed across all six fields to determine the “winningest card”.

The cumulative victories can be used to rank the cards and a category plot illustrates how “winningness” is distributed throughout the deck.

As an aside: looking at the way the scores for each category are distributed is interesting too. Understanding these distributions and the way that each are scaled gives a better feel for whether a score of say 2 in Wisecracks is any good (it is).

The best card in the deck is Iron Man. What is interesting is that Spider-Man has the designation Top Trump (see card), but he’s actually second in terms of wins over all other cards. Head-to-head, Spider-Man beats Iron Man in Skill and Mystique. They draw on Top Trumps Rating. But Iron Man beats Spider-Man on the three remaining fields. So if Iron Man comes up in your hand, you are most likely to defeat your opponent.

At the other end of the “winningest card” plot, the worst card, is Wasp. Followed by Ant Man and Bucky Barnes. There needs to be a terrible card in every Top Trump deck, and Wasp is it. She has pitiful scores in most fields. And can collectively only win 9 out of (6 * 29) = 174 contests. If this card comes up, you are pretty much screwed.

What about draws? It’s true that a draw doesn’t mean losing and the active player gets another turn, so a draw does have some value. To make sure I wasn’t overlooking this with my system of counting victories, I recalculated the values using a Football League points system (3 points for a win, 1 point for a draw and 0 for a loss). The result is the same, with only some minor changes in the ranking.

I went with the first evaluation system in order to simulate the games.

I wrote a first version of the code that would printout what was happening so I could check that the simulation ran OK. Once that was done, it was possible to call the function that runs the game, do this multiple (1 x 10^6) times and record who won (player 1 or player 2) and for how many rounds each game lasted.

A typical printout of a game (first 9 rounds) is shown here. So now I could test out different strategies: What is the best way to win and what is the best way to lose?

Strategy 1: pick your best category and play

If you knew which category was the most likely to win, you could pick that one and just win every game? Well, not quite. If both players take this strategy, then the player that goes first has a slight advantage and wins 57.8% of the time. The games can go on and on, the longest is over 500 rounds. I timed a few rounds and it worked out around 15 s per round. So the longest game would take just over 2 hours.

Strategy 2: pick one category and stick with it

This one requires very little brainpower and suits the disengaged adult: just keep picking the same category. In this scenario, Player 1 just picks strength every time while Player 2 picks their best category. This is a great way to lose. Just 0.02% of games are won using this strategy.

Strategy 3: pick categories at random

The next scenario was to just pick random categories. I set up Player 1 to do this and play against Player 2 picking their best category. This means 0.2% of wins for Player 1. The games are over fairly quickly with the longest of 1 x 10^6 games stretching to 200 rounds.

If both players take this strategy, it results in much longer games (almost 2000 rounds for the longest). The player-goes-first advantage disappears and the wins are split 49.9 to 50.1%.

Strategy 4: pick your worst category

How does all of this compare with selecting the worst category? To look at this I made Player 2 take this strategy, while Player 1 picked the best category. The result was definitive, it is simply not possible for Player 2 to win. Player 1 wins 100% of all 1 x 10^6 games. The games are over in less than 60 rounds, with most being wrapped up in less than 35 rounds. Of course this would require almost as much knowledge of the deck as the winning strategy, but if you are determined to lose then it is the best strategy.

The hand you’re dealt

Head-to-head, the best strategy is to pick your best category (no surprise there), but whether you win or lose depends on the cards you are dealt. I looked at which player is dealt the worst card Wasp and at the outcome. The split of wins for player 1 (58% of games) are with 54% of those, Player 2 stated with Wasp. Being dealt this card is a disadvantage but it is not the kiss of death. This analysis could be extended to look at the outcome if the n worst cards end up in your hand. I’d predict that this would influence the outcome further than just having Wasp.

So there you have it: every last drop of fun squeezed out of a children’s game by computational analysis. At quantixed, we aim to please.

The post title is taken from “Weak Superhero” by Rocket From The Crypt off their debut LP “Paint As A Fragrance” on Headhunter Records

## Pledging My Time II

2016 was the 400 year anniversary of William Shakespeare’s death. Stratford-upon-Avon Rotary Club held the Shakespeare Marathon on the same weekend. Runners had an option of half or full marathon. There were apparently 3.5 K runners. Only 700 of whom were doing the full marathon. The chip results were uploaded last night and can be found here. Similar to my post on the Coventry Half Marathon, I thought I’d quickly analyse the data.

The breakdown of runners by category. M and F are male and female runners under 35 years of age. M35 is 35-45, F55 is 55-65 etc. Only a single runner in the F65 category!

The best time was 02:34:51 by Adam Holland of Notfast. Fastest female runner was 3:14:39 by Josie Hinton of London Heathside.

Congrats to everyone who ran and thanks to the organisers and all the supporters out on the course.

The post title is taken from “Pledging My Time” a track from Blonde on Blonde by Bob Dylan

## Repeat Failure: Crewe Alexandra F.C.

Well, the 2015/2016 season was one to forget for Crewe Alexandra. Relegation to League Two (English football’s 4th tier) was confirmed on 9th April with a 3-0 defeat to local rivals Port Vale. Painful.

Maybe Repeat Failure is a bit strong. Under Dario Gradi, the Railwaymen eventually broke into League One/Championship (the 2nd Tier) where they punched above their weight for 8 seasons. The stats for all league finishes can be downloaded and plotted out to get a sense of Crewe’s fortunes over a century-and-a-bit.

The data are normalised because the number of teams in each league has varied over the years from 16 to 24. There were several years where The Alex finished bottom but there was nowhere to go. You can see the trends that have seen the team promoted and then relegated. It looked inevitable that the team would go down this season.

Now, the reasons why the Alex have done so badly this season are complex, however there is a theme to Crewe’s performances over all of this time. Letting in too many goals. To a non-supporter this might seem utterly obvious – of course you lose a lot if you let in too many goals. But Crewe are incredibly leaky and their goal difference historically is absolutely horrendous. The Alex are currently in 64th place on the all-time table, between West Ham and Portsmouth, with 4242 points – not bad – however our goal difference is -952. That’s minus 952 goals. Only Hartlepool have a worse goal difference (-1042). That’s out of 144 teams. At Gresty Road they’ve scored 3384 and let in 2526. On the road they netted 2135 but let in 3945.

Stats for all teams are here and Crewe data is from here.

See you in League Two for 2016/2017.

The post title is taken from “Repeat Failure” by The Delgados from their Peloton LP.