# CGSS: Introduction to Game Theory

A fundamental problem is over-usage. Usually, nobody wants over-usage to occur. However, on an individual level companies want to maximise their profit while they neglect the social cost. The problem is also known as the Tragedy of the commons which is based on the free-rider dilemma.

A game is defined by three components: players, actions, and payoffs. Solutions concepts are usually in the form of a Nash equilibrium.

“Unless the number of individuals (micro) in a group is quite small,  or unless there is coercion or some other special device to make individuals act in their common interest, rational, self-interested individuals will not act to achieve their common or group interest.” – Mancur Olson(1965)

Players $N = \{1,2,…,n\}$ are a discrete set or continuum population of individuals.

Actions $S = S_1 \times… \times S_n \subset \mathbb{R}^+$ are a compact and bound set, to fulfil Nash’s proof, but some variant are possible to get a fixed point result.

Payoffs $\Pi_i : S \mapsto \mathbb{R} ~ \forall i \in N$ are most often a set of the real space.

Let $\Gamma = \{ N, S, \Pi \}$ be a game with $N = \{1,2,…,n\}$ players. Each $i \in N$ has a strategy set $S_i$ where $S = S_1 \times… \times S_n$ is the set of all possible strategy profiles.

Let $x_i \in S_i$ be a strategy action for $i$ and $x_{-i} \in S_{-i}$ be a strategy profile for all players except $i$. When each $i \in N$ chooses strategy $x_i$ resulting in $x = (x_1, …, x_n)$ then $i$ obtains profit $\Pi_i(x)$.

A strategy profile $x^* \in S$ is a Nash Equilibrium $NE$ if no unilateral deviation in strategy by any single player is profitable for that player, that is, $\Pi_i(x_i^*, x_{-i}^*) > \Pi_i(x_i,x_{-i}^*) ~ \forall x_i \in S_i$ and $i \in N$.

Prisoner’s Dilemma

In the prisoner’s dilemma each of the two players has two action, either to cooperate (C)or to defect(D). The payoffs are defined as  CC=(3,3), CD=(1,4), DC(4,1), and DD(2,2). If player start of with CC, any one player can gain an advantage by defecting. Once defected the other player has no choice but to defect as well to optimize his payoff. The Nash Equilibrium of DD is reached where neither player could optimize his payoff by switching strategy on his own. However, if both were to switch at once, they could get a better result. Since no trust between players exist this outcome is not possible without breaking the rationality assumption.

Public Goods Game

The N-person generalisation of the Prisoner’s dilemma. The players are a finite population of $N$ individuals. Each $i$ is endowed with a budget $B$, common to all players. The action consists of each player $i$ choose some amount  $a_i \in \mathbb{R}^+_0$ to invest in a shared investment account. The collected investments are $a = (a_i)_{i\in N}$. The payoff is $\phi_i(a_i,a_{-i}) = B – a_i + r \cdot \sum_{j\in N} a_j$. For $r < 1$ the optimal decision (i.e. the Nash Equilibrium) is not to invest which results in a payoff of zero $a_i = 0 ~ \forall i \in N \rightarrow \phi_i = 0$. Even though there would be a positive outcome if everybody invests, free-riding would allow some to make more gains. A rational individual would therefore not invest.

Side note:

The lecture on game theory in QPAM is covering the same topic and is therefore not covered in detail.

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