Game theory in Mario Party – Mind Your Decisions

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Nintendo’s Mario Party is a series that is a video game equivalent of board games. Gameplay involves mini-games which might include chance elements like rolling a dice or tactical elements like positioning a character on the right square. These video games consequently include many strategic elements and useful examples of game theory.

In the game Mario Party 2, for instance, there is an interesting mini-game called “Honeycomb Havoc.” I will explain the rules now, but they can also be learned by watching the youtube video below.

The rules, in brief

The game involves 4 players taking turns grabbing items from a tree. Each player hits a slow moving dice block to take either 1 or 2 items at a time. It is not possible to skip a turn and take no items, nor is it possible to take more than 2 items at a time.

The items make the game interesting. The items appear in a predetermined sequence, and they can be one of three choices. An item can be a fruit which is harmless, a coin which is good, or a honeycomb which is lethal.

The point of the game, naturally, is to avoid the honeycombs which are interspersed among the fruits and coins. The winner of the game is the player that avoids the honeycombs and is the last one standing.

 The rules, in action

To see the gameplay and rules in action, check out the following video of the mini-game “Honeycomb Havoc,” courtesy of gametheoryclips:

Youtube video: Honeycomb Havoc in Mario Party 2

What is the strategy of this game?

How can you force an opponent to pick a honeycomb? Or conversely, how can you avoid picking a honeycomb?

Think ahead, reason backwards

The game is definitely one of strategy and calculation, as the intro screen clearly indicates. As a bit of trivia, the CBS show Survivor featured a comparable game known as Thai 21, where two teams alternated turns picking 1, 2, or 3 flags. The team picking the 21st and last flag won the game.

Mathematically, this game is a specific version of a game called Nim. I will spare the gory details of solving Nim and describe how to solve this specific Mario game.

At first the gameplay is complicated with four players and three honeycombs. It is not necessarily possible to calculate what other players will do, and hence there is an interesting twist at the start.

But the game becomes more tractable as it progresses. Suppose you happen to make it to the final round. It is just you and another player, there is some number of fruits and coins ending with a honeycomb at the end. How do you play to win this game?

For illustration, let us take the scenario from the video clip above (the two player starts at about the 1:25 mark).

Yoshi gets to go first against Luigi. The line of items contains 9 items before the honeycomb. How should Yoshi play? Can Yoshi guarantee a win?

It turns out the answer is no. Yoshi should lose if Luigi plays properly, and here is the reason why.

The strategy with two players

A player’s strategy is to force the other to have to take the honeycomb. What circumstances will make this possible?

One obvious answer is when a player faces 1 or 2 items before the honeycomb. In this situation, the player can swoop up all of the fruits or coins and leave just the honeycomb, which the other has to take because skipping is not allowed.

We can label the above situation as a winning position. It is possible, with proper play, to win when facing 1 or 2 items before the honeycomb.

Knowing this, we can reason one step further. A player facing 3 items before the honeycomb, therefore, must be in a losing position. This player can only take 1 or 2 items, which sets up the other player in a winning position.

The logic can again be extended. If 3 items before the honeycomb is losing, then the previous player who forced that position, must have been in a winning position. Hence, a player facing 4 or 5 items before the honeycomb is in a winning position.

A pattern soon emerges that a player facing 3, 6, 9, etc. (multiples of 3) items before the honeycomb is in a losing position. No matter what this player does, the other player can, with proper strategy, force him into a losing position. Or to say it positively, a player facing 1, 2, 4, 5, 7, 8, etc. (not multiples of 3) items before the honeycomb is in a winning position and has the ability to force the other player into a losing position.

Returning to the video, Yoshi faces 9 items before the honeycomb. It is be possible for Luigi in theory to win the game.

Unfortunately that is not how things turn out. The game starts out well. Yoshi takes an item and Luigi responds by taking two, leaving 6 before the honeycomb, putting Yoshi in a losing position.

It is the next round where things fall apart. Yoshi takes two items, leaving 4 items before the honeycomb. At this point, it is evident that Luigi should take only one item to leave Yoshi at 3. But Luigi actually takes two items, leaving Yoshi at 2 items before the honeycomb. Yoshi smartly grabs both of them up, forcing Luigi to grab the honeycomb and lose. But let us not be so critical of Luigi–game theory takes time and practice to master.

The strategy with more than two

The game is not as easy to analyze with three or four players. I don’t think it is possible to calculate a surefire strategy, like in the two person case. But if you have one, do let us know :)

The reason I think this so is the following setup that has two different outcomes. Suppose there are three total players, and suppose you end your turn with 4 items before a honeycomb.

Situation 1: If the following player picks a single item, then the player after that will have to pick one or two. Either way, you start your turn with one or two items before the honeycomb and you can survive.

Situation 2: Suppose instead you end with 4 items and the following player picks two items. In that case, the next player could pick up the remaining pair of items before the honeycomb and you will be forced to lose. Or he might be nice and just take 1 allowing you to live, but the player following you to lose.

What happens depends on how strategic and cutthroat the other players are, and whether you have a pact with anyone before the final round.

Discussion questions

1. It was shown above that having 3, 6, 9, etc (multiples of three) items before the honeycomb was a losing position. What are the analogous losing positions if a player could take 1, 2, or 3 items at a time?

2. What are the losing positions when a player can take 1, 2, 3, …, k items at a time?

3. In Honeycomb Havoc, there is just one line of items, or “heap” that all players are choosing from. How does the game change when there are more heaps? See the following discussion for a mathematical answer.

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