A winning strategy for player is defined to be a strategy such that for every possible strategy of the opponent, the resulting play is winning for player. A game is called determined if one of the players has a winning strategy. Clearly it cannot be the case that both players have winning strategies. One could be tempted to think that, because of the perfect information, one of the players must have a winning strategy. However, because of the infinite duration, one can come up with strange games e.
Let be one of the players. The proof of the above theorem has two parts. Parity games are important because not only can they be won using finite memory strategies, but even memoryless strategies are enough:. Theorem 1. For every parity game, one of the players has a memoryless winning strategy. This essentially boils down to transforming deterministic Muller automata into something called deterministic parity automata.
In a parity automaton, there is a ranking function from states to numbers, and a run is considered accepting if the minimal rank appearing infinitely often is even. This is a special case of the Muller condition, but it turns out to be expressively complete in the following sense:. Theorem 2. For every deterministic Muller automaton, there exists an equivalent deterministic parity automaton.
Consider a game with an -regular winning condition. By Theorem 2, there is a deterministic parity automaton which recognises the language. This is a parity game, with the ranks inherited from the automaton. In a position , the player controlling position chooses an edge in the original game, and the state is updated deterministically according to the transition function of the automaton.
It is not difficult to see that the following conditions are equivalent for every position of the original game and every player : 1. The implication from 1 to 2 crucially uses determinism of the automaton and would fail if a nondeterministic automaton were used under an appropriate definition of a product game. George Berkeley says that a man who believes in no future state has no reason to postpone his own private interest or pleasure to doing his duty 1.
Reciprocity is one way to establish cooperation between rational individuals under this shadow of future 2 , 3 , 4 , 5. WSLS also solves the problem of distinguishability in the sense that it earns a strictly higher average payoff against an unconditional cooperator.
However, it is vulnerable against unconditional defectors. A notable progress in the iterated PD game is the discovery of the zero-determinant ZD strategies This is true even when the co-player has a longer memory or when the strategy is known to the others. When both the players attempt to extort each other using an extortionate ZD strategy, they end up with mutual defection, so an extortionate strategy is hard to evolve as a group 18 , 19 , 20 , The ZD strategies have been studied not only in a well-mixed population but also in structured ones 22 , 23 because of the importance of spatiotemporal dynamics from a statistical-physical viewpoint It has been devised to remedy the problems of TFT by satisfying the following three criteria:.
Efficiency If all the players in the game have adopted this strategy in common, they will reach mutual cooperation with probability one as the implementation error rate e approaches zero. Distinguishability If all the co-players are unconditional cooperators, the expected payoff from this strategy is strictly higher than theirs.
Here, an implementation error also called execution error or mistake refers to an event that a player erroneously takes the opposite action to the prescription of the strategy. Unlike the perception error, it is assumed that all the players, including the one who committed an error, correctly perceive which actions are actually taken. The class of strategies satisfying these three criteria are called successful hereafter.
The first two criteria are especially important because a cooperative Nash equilibrium is formed when efficiency and defensibility criteria are simultaneously satisfied TFT-ATFT is a memory-two strategy, namely, it prescribes its next action depending on the history profile for previous two rounds.
Otherwise, it behaves as ATFT. If mutual cooperation is reached, or if the co-player unilaterally defects twice in a row, it is time to go back to TFT. Thus, when the player erroneously deviates from TFT, the ATFT part is activated for a while to correct the error, whereby mutual cooperation can be made robust in a noisy environment without violating defensibility. Regarding efficiency, we mention that perception error can also be corrected if it occurs with a much longer time scale than implementation errors Successful strategies exist not only for the iterated PD game but also for an iterated public-goods PG game The payoff matrix of the three-person PG game is given as follows:.
This is a generalization of the iterated PD game to a three-person case. For the iterated three-person PG game, it has been found that at least successful strategies exist in the memory-three strategy space 27 and that no such strategy exists if the memory length is less than three. The purpose of this paper is to interpret the successful strategies by representing them as automata.
However, a strategy may also be defined as an automaton 28 , i. The paper is organized as follows: In the next section, we present an algorithm to convert a history-based representation to a state-based one. Then, its applications to some successful strategies will be demonstrated. We discuss possible interpretations for the resulting internal states and summarize this work in the last section. In this section, we show how a history-based strategy can be converted to a state-based representation.
In general, history-based strategies may be regarded as a subset of state-based ones because one may also regard the history profile over the previous m rounds as an internal state. In this naive reinterpretation, the number of states i. Note that this graph does not include transitions caused by implementation error.
An example of such a graph is shown in Fig. Because TFT is a memory-one strategy for a two-players game, it has four nodes, labelled by cc , cd , dc , and dd , respectively. Suppose that the current history profile is cc. Although this representation fully defines the strategy, it has redundancy. For instance, it is obvious that TFT can also be represented by a graph with two states as shown in Fig. In case of TFT, it is straightforward to construct the graph in Fig.
However, it suddenly becomes complicated when the memory length gets longer because the number of nodes grows exponentially. Thus, the question is how to simplify a naive representation systematically by minimizing the number of states. This is known as deterministic-finite-automaton DFA minimization in automata theory Specifically, we use the following algorithm:. Increase k by one. In our context, an input means an action tuple of the co-players.
In short, we regard two states as identical when they lead to the same future. The algorithm always terminates after a finite number of steps, and the final result is uniquely determined irrespective of the order of choosing node pairs. If we apply this algorithm to Fig.
The opposite conversion is not always possible. For example, one needs an infinitely long memory to describe the behaviour of Contrite TFT CTFT 30 in the history-based representation 31 , whereas its state-based version needs only four states Fig. It greatly simplifies the graphs, especially when the memory length is long. Here, we note that the transitions in Fig. In other words, the minimized automaton generally loses some of information about erroneous actions while it reproduces the deterministic actions prescribed by the strategy.
In order to fully keep the information of the original history-based representation, one needs to start from the transition graph that has outgoing links corresponding to erroneous actions as well. An example of such automaton representation will be shown in Fig. In general, we should choose one of the representations depending on our purpose. While information loss is caused by ignoring error, the converted automaton representation usually has a smaller number of states, which is helpful in interpreting the strategy.
On the other hand, the full automaton representation keeps all the information of the strategy, making it possible to reconstruct the history-based representation. In this paper, we mainly take the former approach because our main objective is to better interpret the strategies. Each node is labelled by a history profile, which is a 2-tuple composed of the last actions of the two players in this memory-one strategy. A history profile may also be regarded as an internal state of the focal player in this naive representation.
If the co-player cooperates defects , the internal state becomes 0 1 , and the focal player chooses an action based on this state. The colour of each node indicates the action prescribed at each state: Blue and red mean cooperation and defection, respectively. DFA minimization.
Conversion of history-based representation to state-based one. Each of these strategies, which generally has 16 nodes as a memory-two strategy, is reduced to an automaton with three internal states by the DFA minimization. As in Fig. We have suppressed the action tuples assigned to the links in the history-based representation for better visibility.
Each node represents a history profile of the two previous rounds, thus the graph has 16 nodes in total. The green dashed rectangle shows the strongly connected component responsible for the TFT behaviour. The state changes according to the 2-tuple of actions attached to each link. The four dashed rectangles in red and blue in a correspond to the four nodes in b. They are 4 and 12 in binary, and we have chosen the former one to denote the super-node.
Likewise, the label of each super-node in b originates from the minimum index of its constituent nodes in a. This representation is a simplification of a , thus error-induced transitions are not taken into account. This representation is equivalent to the original history-based representation in a. Let us consider the iterated PD game between two players, say, Alice and Bob. Alice normally behaves as a TFT player, and this behaviour is described by the strongly connected component indicated by the green dashed rectangle in Fig.
However, when she erroneously defects from mutual cooperation, she switches her behaviour to ATFT. The history profile jumps from cccc to cdcc by this error, and then Alice should defect once again as an ATFT player. The DFA minimization algorithm simplifies the graph to a great extent as shown in Fig. Although this automaton representation is meant to ignore error as we have mentioned, we depict a dashed arrow in Fig.
This transition is the most important to understand how efficiency is satisfied because it is the only erroneous transition occurring with probability of O e when two players adopt TFT-ATFT. For the sake of completeness, Fig. We can fully reconstruct the original history-based representation of Fig. The colour and the number of each node are depicted in the same way as in Fig. It has been proved for this game that successful strategies are possible only when the memory length is greater than two.
However, it is instructive to begin with partially successful strategies PS2 27 , which are memory-two strategies with defensibility, distinguishability, and partial efficiency. For example, TFT is partially efficient. The history-based representation needs 64 nodes, which makes it difficult to interpret how the strategy works by visual inspection Fig. On the other hand, its state-based representation needs only 6 nodes as demonstrated in Fig.
We can interpret the nodes in Fig. The meaning is obvious: She distrusts Bob. It means that she is in despair because they are trapped in mutual defection. In plain words, therefore, we could say that Alice wants to make an apology at this state. This loop thus provides distinguishability for her PS2.
One of the simplest and one of the most complex strategies are depicted in a , b respectively. The labels and the colours are given in the same way as in Fig. The dashed orange arrows indicate erroneous actions occurring while recovering mutual cooperation from one- and two-bit errors.
The Greek letters correspond to the transitions shown in Fig. One of the simplest is depicted in Fig. Its similarity to Fig. As shown in these automata representation, they show overall similar structures with sharing key mechanisms. Due to this split, it takes one more step to despair when one of the co-players defects. It means that the following recovery path is possible even if Bob defects twice in a row:.
They are transient nodes with no incoming links, which are reachable only by error. Paths for recovering mutual cooperation from one- and two-bit errors. At each node, we have specified which history profile it represents, together with the corresponding internal state in the state-based representation see the node labels in Fig.
The label of an internal state is written in blue red if c d is prescribed at the state. This figure contains all the possibilities up to permutation of the players. The orange Greek letters correspond to those in Fig. In fact, these additional four states are needed to make this strategy tolerant against two-bit error, i. Such tolerance is a necessary condition for full efficiency in this three-person game When Alice, Bob, and Charlie have adopted this FUSS in common, we can show that the players recover cooperation from every possible type of one- and two-bit error by enumerating all the possible cases:.
We have already seen from Eq. In Fig. In the second case, Bob first defects in error from full cooperation. Charlie is supposed to punish Bob by choosing d at the next round, but he mistakenly chooses c instead. After provoking Bob and Charlie by choosing d , Alice wants to make an apology, and Bob and Charlie want to punish her provocation.
For every parity game, one of the players has a memoryless winning strategy. This essentially boils down to transforming deterministic Muller automata into something called deterministic parity automata. In a parity automaton, there is a ranking function from states to numbers, and a run is considered accepting if the minimal rank appearing infinitely often is even. This is a special case of the Muller condition, but it turns out to be expressively complete in the following sense:.
Theorem 2. For every deterministic Muller automaton, there exists an equivalent deterministic parity automaton. Consider a game with an -regular winning condition. By Theorem 2, there is a deterministic parity automaton which recognises the language. This is a parity game, with the ranks inherited from the automaton.
In a position , the player controlling position chooses an edge in the original game, and the state is updated deterministically according to the transition function of the automaton. It is not difficult to see that the following conditions are equivalent for every position of the original game and every player : 1. The implication from 1 to 2 crucially uses determinism of the automaton and would fail if a nondeterministic automaton were used under an appropriate definition of a product game.
Since the product game is a parity game, for every position , condition 2 must hold for either player 0 or 1; furthermore, a positional strategy in the product game corresponds to a finite memory strategy in the original game, where the memory is the states of the automaton.
We consider games played on graphs equipped with costs on edges, and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Save my name, email, and website in this browser for the next time I comment. Weighted automata Syntactic weighted automata Finite dimension — decidable problems Finite dimension — undecidable problems 4.
Distance automata 5. Tree-walking automata 6. Transducers One-way transducers Two-way transducers Register transducers Two-way and register are the same 7. Learning automata 8. Automata with infinite alphabets 9. The finite state stochastic automaton is first considered in a game with nature, and conditions under which the automaton's winnings reach the Von Neumann value of the game are established. Next, two stochastic automata with an arbitrary number of states for each are considered in a game, the game matrix being specified.
Performance of the automata for various conditions on the elements of the game matrix is considered. In a comparison of performance with deterministic automata, it is established that, for performance comparable to that of the finite state stochastic automaton, the deterministic automaton needs an infinite number of states.
Finally, some games are simulated on a computer which verifies the general analysis and further sheds light on the details of the game. Article :. Date of Publication: April
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