Definitions

In the framework described here, a finite state machine is used to represent interactions between deterministic memory-n strategies. The finite state machines (sometimes referred to as finite automata) are dynamic systems that only change their behavior at discrete moments under consideration. The system consists of a finite set of internal states and a transition function. The transition function determines a subsequent system state as a function of the current state plus input. In [93], a formal definition of a finite state was given as:

Definition 4.1 A finite-state machine (finite automata) consists of a 5-tuple (Q, ∑, q0, δ, A),

where

Q is a finite set (whose elements we will think of as states),

∑ is a finite alphabet set of input symbols,

q0 ∈ Q (the initial state),

A ⊆ Q (the set of accepting states), and

δ is a function from Q × ∑ to Q (the transition function).

For any element q of Q and any symbol a∈∑, δ(q,a) is interpreted as the state to which the

finite state machine moves if it is in state q and receives the input a.

A finite state machine can be represented in terms of a state transition diagram, described as:

Definition 4.2 A finite state machine (Q, ∑, q0, δ, A) can be represented as a state

transition diagram {V, E}, where

V is a finite set of vertices, and each vertex represents a state in set Q,

and (qi, qj) ∈ E if δ(qi,a) = qj, where a∈∑.

The diagram is a directed graph. The initial state q0 and the set of accepting states A are marked with specific notation. Each edge (qi, qj) is marked with an input symbol a, indicating that δ(qi,a) = qj.

Regarding the memory-n strategies, cooperative or defection moves are determined by the historical moves of two players. A memory-n strategy determines a move in correspondence to moves made during the last n round, which is represented as a history string (HS). Formally defined,

Definition 4.3 Let HS be the history string of Si in its interaction with Sj, where Si and Sj

are memory-n strategies. Then,

HS=h2nh2n-1…,hk+1hk,…h2h1.

The even and odd indices indicate the respective moves of a player and an opponent.

Here hk∈{C, D}. C represents a cooperative move and D represents a defection move.

Note that in this case, HS’ (the history string of Sj,) would be

HS’=h2n-1h2n…,hkhk+1,…h1h2.

In other words, the two strategies have different history strings. In all, there are 2²ⁿ distinct history strings for a memory-n deterministic strategy. For convenience, the

following formal definition of the order of history strings for a memory-n strategy is offered.

Definition 4.4 {HSi/N | 0≤ i <N and N=2²ⁿ } is the set of all possible history strings for a memory-n strategy, where HSi/N is the ith string in the lexicographic order:

HS0/N = CC…CC,

HS1/N = CC…CD,

HS2/N = CC…DC,

HS3/N = CC…DD,

…

HSN-1/N = DD…DD,

where |HS0/N|=|HS1/N|=…=|HSN-1/N|=2n.

A memory-n strategy is described in terms of the moves that are made according to their history strings.

Definition 4.5. A memory-n deterministic strategy S is represented as:

where Pk∈{C, D}, and N=2²ⁿ. Pk represents the move S corresponding to history string HSk/N.

A memory-1 strategy is expressed as (P0, P1, P2, P3), where P0, P1, P2, P3 indicate moves when the results of the preceding round are CC, CD, DC, and DD, respectively.

For example, strategy (C D C D) indicates a cooperative move if the preceding round is either CC or DC; a defection move is indicated in all other cases.

After finishing a round, the history strings of both strategies are updated by deleting the two leftmost characters, and adding the result of the next round to the right. For example, if a current history string is CCDDCD and the last round is DC, then the history string becomes DDCDDC. Interactions between player strategies can be viewed as history string transitions. Interactions between two strategies can be expressed as a finite state machine, where each state represents a specific history string, and where transitions between states are dependent upon transitions between the strategies’ history strings.

Proposition 4.1 Interactions between two memory-n deterministic strategies can be

represented as a finite state machine.

Proof:

Assume two memory-n deterministic strategies, Si and Sj, represented as

Si = (P0, P1, …PN-2, PN-1), and

Sj = (P0’, P1’, …PN-2’, PN-1’),

where N=2²ⁿ and Pk and Pk’ represent the moves of Si and Sj corresponding to history string HSk/N.

Let FSM(Si|Sj) be a finite state machine whose state transition diagram is:

D(Si|Sj) = {V, E}.

The set of vertices V and the set of edges E are defined as

V = {Vk | Vk is a vertex corresponding to history string HSk/N, 0≤k<N, N=2²ⁿ }, and

E = {(Vu, Vv) | Vu, Vv ∈ V,

where Si’s history string may transit from HSu/N to HSv/N when interacting with Sj, and 0≤u,v<N }

In V, Vk is the vertex corresponding to history string HSk/N. Since there are N vertices, all possible history string permutations are contained.

In E, the edge (Vu, Vv) indicates that the history string of Si will transit from HSu/N to

Interactions between Si and Sj can be represented by D(Si|Sj) without loss of information. Thus, interactions between two memory-n deterministic strategies can be represented as a finite state machine.

Note that the history strings in V and E refer to the history strings of Si, which are different from those of Sj. D(Si|Sj) represents the interaction between Si and Sj from Si’s perspective. The state transition diagram that represents the interaction between Si and Sj

from Sj’s perspective is denoted as D(Sj|Si).

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Definition 4.6 The finite state machine that represents interactions between two memory-n

deterministic strategies Si and Sj from Si’s perspective is denoted as FSM(Si|Sj). Its state transition diagram is D(Si|Sj). Thus,

D(Si|Sj) = {V, E},

where

V={Vk | Vk is a vertex corresponding to history string HSk/N, 0≤k<N, and N=2²ⁿ}, and

E={(Vu, Vv) | Vu, Vv∈V, where

Si’s history string may transit from HSu/N to HSv/N when interacting with Sj for 0≤u<N }.

In common finite state machines, state transitions depend on the current state and input symbols. However, the input symbol in FSM(Si|Sj) is ignored; another way of saying this is that there is only one kind of input symbol, so the state transition depends only on the current state.

To give a simple example, if Si = (C, D, C, D) and Sj = (C, D, D, C), the interactions are represented as a finite state machine FSM(Si|Sj), whose state transition diagram D(Si|Sj) is shown as Figure 4.1. If Si and Sj both made defection moves in the previous round, Si

will make another defection move and Sj will make a cooperative move, resulting in a transition from state DD to state DC.

CC CD DC DD

Figure 4.1: The Interaction between Strategies (C, D, C, D) and (C, D, D, C).