# Balancing a game, or proving the imbalance of a game

Consider a real-time game with two teams $T_1$ and $T_2$ fighting each other, each team composed of $n$ players $p_1^t,\dots,p_n^t$, where $p_i^t$ denote the $i$th player of the team $T_t$. Each player has a list of skills $S_1(p_i^t),\dots,S_k(p_i^t)$. The size $k$ of the list may change from a player to another. Each skill may or may not have a cooldown, in other words, it may be used only once every $c(S_k(p_i^t))$ seconds, where $c(S)$ is a function that returns the cooldown for skill $S$. Each skill has an effect on the game state and each skill may be successfully applied to may fail with some probability (dependent on the game state).

There are two particular game states $W_1$ and $W_2$, which represent a winning state for team $t_1$ and a winning state for team $t_2$. The winning team is the first one who puts the game state in the corresponding winning state.

Maybe some of you recognized the scheme here, games like world of warcraft are similar to the game I just described.

For example, if you consider a duel between two players, $t_1=\{p_1^1\}$ and $t_2=\{p_1^2\}$. If these two players have a different list of skills, and the goal is to reduce the opponent health points to zero before he does reduce your health points to zero, then the game may not be balanced.

My goal is to find a method to show that such games (duels, $2$ vs $2$, $3$ vs $3$, ..., $n$ vs $n$) are either balanced or imbalanced. And if they are imbalanced to propose a method to balance them.

We can sketch a battle: consider a duel, player $A$ will use skills indexed by $a_1,\dots,a_i$ while player $B$ will use skills indexed by $b_1,\dots,b_j$. We assume that $W_A$ or $W_B$ is attained during the application of these strategies. If $W_A$ is attained, then strategy $A$ is a winning strategy against $B$.

Considering $A$ and $B$ always do their best to win the game, I would like to show that the probability $P(W_A)=P(W_B)=0.5$.

I would also like to find a way to consider the difficulty in applying a given strategy.

Right now, I thought about some minmax approach with stochastic state transitions. But I am very unsure how to apply it.

• What does it mean that the game is balanced in this case? – Student tea Jan 3 '15 at 11:49
• The game is not deterministic. Some skills have an effect with a magnitude following a random distribution. That said, balanced would mean that the expectation of winning a game is 50% – davcha Jan 3 '15 at 14:22