Minibattle

The **Minibattle** mechanism is an implementation of the game Matching Pennies. As a game of Minibattle is initiated, the player landing on the Minibattle block will become **the Host of the game**, while **all opponents are the guessers**.

Everyone selects a side of the coin: the **Sun (golden)** or the **Moon (silver)**. When the coin is revealed, guessers who predict the correct side the Host has chosen **will win a fixed amount of money from the Host** multiplied by the active Minibattle Multiplier. Guessers who failed to predict the correct side **will lose money** instead; they will lose the money they would have won multiplied by the Minibattle Payout Increase character stat of the Host's character.

### Minibattle Multiplier

Every time a player lands on Minibattle and plays the minigame, the Minibattle Multiplier **doubles**. The passive Minibattle Multiplier will always be of powers of two. Additionally, some effects and cards may temporarily boost the multiplier. An active Minibattle Battlefield Daily Buff will make the Minibattle Multiplier start at 2x; otherwise, it starts at 1x (no multiplier). The stacking multiplier caps at 64x and does not increase beyond that.

Some effects may increase the multiplier temporarily. They can be found in the Strategy section below.

The Minibattle Multiplier is a factor affecting the amount of money Where's My Money? steals.

## Strategy

### Skill Cards

* **Pro Gambler** lets you move to the Minibattle block and play the minigame if you are not already on the block, **with a higher payoff** for the round if you win.
* **Big Match** lets you play the minigame (even if you are already on this block, unlike Pro Gambler) **with a higher multiplier** for the round.
* **Where's My Money?** **steals wealth** from the last Minibattle initiator if it was an opponent, with its amount based on the current multiplier. It is a powerful tool in a mirror gambler deck, as the multiplier can get very high from the constant hits.
* **It's a Trap!** sets up a penalty chance and **steals parts of the wealth** of the landing player, and in addition **move them to Jail** instead of triggering the minigame. This is a tactical card that can protect you from being bankrupt as a result of an opponent winning the Minibattle; especially in the endgame where you have loaned (and cannot risk it) and the multiplier is scary.

### Dice Effects

* **Minibattle Duelist** may trigger and **double the multiplier** for the turn.

## Solving the game

#### Two players (One guesser)

The game can be written in a payoff matrix (as depicted below). Each cell of the matrix shows the two players' payoffs, with the Host's payout written first. (x is the increased payout of the host (1 + minibattle payout increase rate); applying the multiplier and base money calculations will be left as an exercise for the reader.)

This game has no pair of pure strategies such that both players would have a best response to a best response. Instead, we must solve for a mixed strategy equilibrium:

**Solving for the Host's Mixed Strategy**

```
EU sun = EU moon ... (1)
EU sun = f(σ sun) ... (2)
EU moon = f(σ sun) ... (3)

(2): EU sun = σ sun(-1) + (1 - σ sun)(x)
(3): EU moon = σ sun(x) + (1 - σ sun)(-1)

(1): σ sun(-1) + (1 - σ sun)(x) = σ sun(x) + (1 - σ sun)(-1)
 -σ sun + x - x σ sun = x σ sun - 1 + σ sun
 2x σ sun + 2 σ sun = x + 1
 σ sun = 1 / 2

EU sun + EU moon = (1 / 2)(-1) + (1 - (1 / 2))(x) + (1 / 2)(x) + (1 - (1 / 2))(-1)
 = x - 1

```

**Solving for the Guesser's Mixed Strategy**

```
EU sun = EU moon ... (1)
EU sun = f(σ sun) ... (2)
EU moon = f(σ sun) ... (3)

(2): EU sun = σ sun(1) + (1 - σ sun)(-x)
(3): EU moon = σ sun(-x) + (1 - σ sun)(1)

(1): σ sun(1) + (1 - σ sun)(-x) = σ sun(-x) + (1 - σ sun)(1)
 σ sun - x + x σ sun = -x σ sun + 1 - σ sun
 2 σ sun = -2x σ sun + 1 + x
 (2 + 2x) σ sun = 1 + x
 σ sun = 1 / 2

EU sun + EU moon = (1 / 2)(1) + (1 - (1 / 2))(-x) + (1 / 2)(-x) + (1 - (1 / 2))(1)
                 = 1 - x

```

Therefore the best strategy for both players is to pick each side with a **50% chance**. The host has a positive expected payout while the guessers expect a negative payout.