I’d typically assume that after most card games, the cards become at least semi-ordered in some way, necessitating shuffling. However, after a standard game of war, I can’t quite figure out how the arrangement would become less random, since the winning and losing card stay together. If they’re indeed mathematically “less random,” after the game, why?
In: 312
Am I the only person who has never heard of this card game before…?
Unless you have a very particular way of ordering the cards after a war, I would think two cards of the same rank are less likely than a randomly shuffled deck of cards to end up on the bottom.
Fun fact: There are so many permutations of the order of a deck of cards that it’s almost statistically impossible for anyone to have ever held a deck with the exact same order of cards as any shuffled deck you’ve held.
Edit: And they probably won’t for a very very long time.
Entirely depends on your definition of random. Off the top of my head:
* When comparing neighbouring cards, you should find next to no pairs.
* When looking at top cards – say, aces+kings+queens – you would find them more frequently in top side of the deck – games end with losing deck having no good cards.
* If the order of winning/losing card is maintained, that’s a dead giveaway – and unlike other poster said, we did do it religiously. Nothing like realising you’re about to lose an ace because of what you did two cycles ago.
The question is – what do *you* consider random. Is it ability to deduce war was played? Not very random. Is it difficulty to shuffle into no visible bias? Then it’s almost random. Technically, every combination cards can take up in a deck is equally random, so it becomes more of a question of “what things would you check to determine if it’s a random shuffle?” You determined one: you’d look for good (which ones precisely would depend on the game) cards at the bottom and clumped up. To this test, war is fairly random. It introduces other biases to the deck – but they might not be ones you care about.
It is mathematically a bit less random, but here’s precisely why:
In case of a draw – cards of the same value are pinned against each other. Another two cards are placed on top of first and second one. When collected usually you will pick one pile and then the next. This means it’s a tiny bit more likely that for every card of some value there’s another card of the exact same value, exactly 2 cards away in the deck (or more, if it was draw into draw).
I can’t prove this yet without some more thinking, but I think the expected value nature of the game will lead to a derandomizing. By this I mean:
13 values possible (2-10, J, Q, K, A). Imagine dividing the deck in thirds.
2,3,4,5 are very likely to ‘move’ on any given turn.
J, Q, K, A are very unlikely to move, and will act as magnets round after round.
6,7,8,9 are basically wildcards.
To me, I think what I’m getting at is that in each subsequent round a magnet card has a low *or* neutral card next to it. If it wins again, and the neutral card survives, it now has two low/neutrals next to it. Sure, it may lose the low card the next flip, but over the long run the power of the magnets should pretty well disperse them through the deck, surrounded by their neutral but not terrible colleagues, with the low cards that are traded back and forth endlessly sorting to the end.
So decks should show some mild movement from a random arrangement to stacks of thirds (or quartiles, if you want to put them in sets of 3 values each instead of 4 as I did above) roughly ordered high to low. How roughly depends entirely on the original random arrangement.
Edit: this assumes the cards are collected consistently as someone else called out. I tend to collect consistently, others may not.
One point folks are missing is that, although the deck may still be random, and so alright to hand to someone else for their game of war without shuffling, it may still not be random from the perspective of the current players.
Although I don’t remember exactly how war plays out, I assume after a game you could theoretically memorize which cards went back on top or bottom. This could help you figure out who has better odds at winning and so help you place your bets and increase your chance of making money, or bragging rights if no money is at stake. You could even rig the deck by cleaning things up creatively. Even without trying to cheat, you might accidentally recognize a sequence of cards and thus accidentally spoil the game. I think I’ve seen this happen before.
Basically, randomness can be subjective.
A deck can go from random to non-random without ever adjusting the order, if you just decide to scan and memorize it.
A very real change occurs. The top card goes from a 1/52 chance of being an ace of spades to either a 0% or 100% chance, as one example.
It would have exactly the same level of “randomness”, because the entire game is determined at the point of the initial deal, and there aren’t specific patterns in the cards that are played.
In some games you choose which cards to play, like poker. So in the final deck of a poker game, you may expect certain patterns to emerge, like cards of the same face value being together and straights together. You could consider this “less” random, because it would be more easy to make predictions based on what cards you see.
In War, you don’t have agency. So if two games are played such that the cards happened to be dealt the exact same, then the output would be the exact same. There would be no specific patterns in the output, either. So, it’s the exact same “randomness” as the original deck, given some cards, you have no better than a random chance at being able to guess the next card.
I would think high cards would tend towards having sequential descending cards following them in the deck over time, assuming that the winner adds the hand back to their deck with the winning card in a position to be played first.
This is because higher cards are less likely to change hands, and can only be captured by high cards themselves.
So say you play a Queen and are lucky enough to capture a Jack. On future rounds, that Queen might capture a 4 and a 6, but when that 4 and 6 trot out into the field of battle they’re more likely to leave your deck than the Jack. So the Jack would tend to move closer to the Queen over time.
Probability quantifies uncertainty. When we say a system is “more random”, we mean that we have less knowledge about its state. Shuffling resets our knowledge of the system to maximum randomness, i.e. a situation in which every player has no reason to believe any permutation of the deck is any more likely than any other. (The quantity that represents the “randomness” of a distribution is called “entropy”, and a uniform distribution maximises entropy for a given set of possible states.)
Something to understand is that probability is in a sense subjective. A probability distribution represents a state of knowledge. For example, if I flip a coin and ask you what the probability is that it’s heads, you would answer 50%. But as I can see the outcome of the coin flip, from my perspective the probability can only be either 100% or 0%.
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