# A Rough Perspective on Information in Extensive Form Games

## Abstract

In game theory imperfect and incomplete information have been intensively addressed. In extensive form games a player faces imperfect information when it cannot identify the decision node it is presently located at. The player is only aware of an information set consisting of more than one node. A player faces incomplete information when it is not aware of, e.g., preferences or payoffs of its opponents. Rough set theory is a prime method addressing missing and contradicting information in decision tables where a set of variables induces a decision. In particular, rough set theory provides a means by which records with identical variable values lead to different, contradicting decisions. To indicate such situations, these records are assigned to the boundaries of all possible decisions. Obviously, both situations, games with imperfect or incomplete information and rough decision tables are similar with respect to their characteristics and challenges regarding a lack of information. Hence, a discussion of their relationship could be mutually beneficial. Therefore, the objective of our paper is to provide a rough set perspective on extensive form games with imperfect and incomplete information.

### Keywords

Rough set theory Game theory Extensive form games Imperfect information Incomplete information## 1 Introduction

Game theory is widely applied in a diverse range of areas. Its objective is to optimize payoffs in situations with two or more players. Important fields of application of game theory are in economics and social science where it has been used to investigate and understand human behavior [8]. However, in the past decades, it suitability for economic analysis has been questioned. It has been observed that important preassumptions of game theory do not match with human behavior (basically assuming a homo economicus). Therefore, a new field, behavioral economics [1], has emerged where experiments are performed to understand human behavior.

Game theory is also applied to a wide range of ‘technical’ fields, i.e., where it is irrelevant how humans behave. These fields include engineering and computer science [4] and many others. Its ‘technical’ applications as an optimization method also include rough sets. In game theoretic rough sets [6, 7], game theory is applied to reduce the boundaries by moving selected objects from there to positive and negative regions. So, in game theoretic rough sets, rough sets are not integrated into game theory in a sense of rough game theory but applied to optimize rough set approximations. In contrast to this, Xu and Yao [9], for example, integrated rough sets into game theory by developing a rough payoff matrix derived from rough variables.

Since practically any real life situation is characterized by a lack of information intensive attention has been given to games dealing with imperfect and incomplete information. Often the terms (im)perfect and (in)complete are not precisely defined and used interchangeably addressing any lack of information in a systems. For example, at EconPort [3] perfect information is defined as follows: “By perfect information we mean that anything that may impact a buyer or seller’s decision making process is known and understood.” Hence, imperfect information is given when information is incorrect, incomplete or missing.

However, in game theory the terms imperfect information and incomplete information have different meanings. Imperfect information can be observed in extensive form games, i.e., games in tree forms, when a player does not know its present position in the tree at all times. Such situations occur when it is not aware of all previous decisions taken by the other players. In contrast to this, incomplete information refers to games where one player has only limited information about the preferences, payoffs etc. of the other players. So, imperfect information is associated with past actions of a player’s opponents, while incomplete information is linked to future actions of a player’s opponents.

Imperfect and incomplete information have been extensively addressed in game theory and have led to several refinements of the equilibria in games (see, e.g., Bonanno [2] for a good introduction). The relationship between imperfect and incomplete information has also raised great attention. When some assumptions are made about the preferences of the players and about probabilities, a game with incomplete information can be transformed into a game with imperfect information (Harsanyi transformation [5]).

Rough set theory addresses missing and contradicting information in decision tables. For example, two objects are indiscernible, i.e., they have identical attribute values. Sometimes these objects lead to different decisions. Reasons may include that a crucial attribute is missing or that the data recorded are inconsistent. To indicate this, such objects are assigned to the boundaries of all possible decisions. When objects with identical attribute values lead to identical decisions, they are regarded as sure objects and are assigned to the lower approximation of the respective decision.

So, regarding the emphasis to deal with information, game theory and rough sets seem to be rather similar. However, little attention has been directed to the relationship of imperfect and incomplete information in game theory and rough sets. Therefore, the objective of the paper is to discuss the relationship of imperfect and incomplete information in extensive form games and rough sets. We limit our presentation on a rough set perspective on games and only address some very key concepts of game theory. So basically, we provide a ‘rough’ rough perspective on extensive form games in our paper.

The remainder of the paper is organized as follows. In Sect. 2, we discuss imperfect information. In the next section, we deal with incomplete information. In Sect. 4, we merge imperfect and incomplete information into one rough decision table and develop a rough payoff matrix. The paper concludes with a summary in Sect. 5.

## 2 Imperfect Information

### 2.1 Imperfect Information in Extensive Form Games

In extensive form games imperfect information is defined when a player does not always know at which decision node it is located. For example see Fig. 1 that shows an extensive form game with two players. Player A, indicated by coarse dotted lines, has two decision nodes (\(A_1\) and \(A_2\)) while for Player B, indicated by solid lines, there are three decision nodes (\(B_1\), \(B_2\) and \(B_3\)). The results are depicted as circled \(C_i\) with \(C_i=(c_{Ai}, c_{Bi})\) the payoffs for Players A and B, respectively.

A strategy is a predefined set of actions that determines how a player will decide at any information set of a game, i.e., it is a complete guide to action. Player A has perfect information, i.e., each decision node forms an information set. Hence, each of its strategies comprises of two predefined actions (for node \(A_1\) with three possible decisions (up: \(\uparrow \) towards \(B_1\), right: \(\rightarrow \) towards \(B_2\) or down: \(\downarrow \) towards \(B_3\)) and for \(A_2\) with two possible decisions (up: \(\uparrow \) towards \(C_1\) or down: \(\downarrow \) towards \(C_2\))). Therefore, Player A has a total of \(6=3 \cdot 2\) strategies. In contrast to this, Player B faces imperfect information with two information sets only for three nodes. At each information set it can go up (\(\uparrow \)) or down (\(\downarrow \)) which leads to \(4=2 \cdot 2\) strategies. Moving up at \(I_{B1}\) leads to \(A_2\) or \(C_4\), moving down to \(C_3\) or \(C_5\) depending on the node it is.

Obviously, we obtain different games if a player has perfect information or only imperfect information. In the case of perfect information, we get a game as depicted in Table 1. The left matrix in Table 1 shows the full redundant matrix while the right matrix shows the minimum matrix. The arrows indicate the decisions the players take at a node. E.g., strategy \(b_2\,=\,\uparrow \uparrow \downarrow \) for Player B means that it moves up at \(B_1\) and \(B_2\), and it moves down when it is at \(B_3\). In the minimum matrix, a star \(*\) indicates that any decision taken at a particular node leads to the same result.

Game with perfect information

Game with imperfect information

### 2.2 Imperfect Information and Rough Sets

Imperfect information: rough strategies of Player B

Rough strategy | Player B | Player A | Payoffs | |
---|---|---|---|---|

Action | at node | Action at \(A_1\) | ||

\(b_1\) | \(\uparrow \) | \(B_1\) | \(\uparrow \) | \(C_1\) or \(C_2\) |

\(b_2\) | \(\uparrow \) | \(B_2\) | \(\rightarrow \) | \(C_4\) |

\(b_3\) | \(\downarrow \) | \(B_1\) | \(\uparrow \) | \(C_3\) |

\(b_4\) | \(\downarrow \) | \(B_2\) | \(\rightarrow \) | \(C_5\) |

\(b_5\) | \(\uparrow \) | \(B_3\) | \(\downarrow \) | \(C_6\) |

\(b_6\) | \(\downarrow \) | \(B_3\) | \(\downarrow \) | \(C_7\) |

Let us first discuss the strategies \(b_5\) and \(b_6\) of Player B. Since it is aware that it is at node \(B_3\), it can distinguish between the strategies and select the strategy that optimizes its payoff. For \(c_{B6} \succ c_{B7}\) it would choose going up (\(\uparrow \)) and for \(c_{B6} \prec c_{B7}\) it would go down (\(\downarrow \)) while it would be indifferent for \(c_{B6} \sim c_{B7}\) (\(*\)). Hence, in rough set terms we suggest to assign the strategies \(b_5\) and \(b_6\) to lower approximations.

In contrast to the above, due to imperfect information, it cannot distinguish whether it is at node \(B_1\) or at node \(B_2\). These nodes are indiscernible for the player. If it decides to move up (\(\uparrow \)) and happens to be at node \(B_1\) the payoffs \(C_1\) will be obtained. If it happens to be at node \(B_2\) it is heading towards \(C_4\). When it decides to move down (\(\downarrow \)) it ends up at \(C_3\) if it happens to be at node \(B_1\) and at \(C_5\) if it is at node \(B_2\). Hence, in rough set terms we would assign the strategies \(b_1\), \(b_2\), \(b_3\) and \(b_4\) to boundaries.

Irrelevant Boundaries. Assuming that \(c_{A6}, c_{A7} \succ c_{A1}, c_{A2}, c_{A3}, c_{A4}, c_{A5}\), Player A will select to go down (\(\downarrow \)) at node \(A_1\) which leads to node \(B_3\) of Player B. Node \(B_3\) is identifiable for Player B. Hence, Player B is not challenged by any imperfect information since it will never be at \(I_{B1} = \{B_1, B_2\}\). There are still boundaries in the game but they are irrelevant (dominated).

Weak Boundaries. For \(c_{A6}, c_{A7} \prec c_{A1}, c_{A2}, c_{A3}, c_{A4}, c_{A5}\) Player A will go up (\(\uparrow \)) or to the right (\(\rightarrow \)) at node \(A_1\). For our discussion, it is irrelevant where it actually goes since both pathes lead to the same information set \(\{B_1, B_2\}\) for Player B. Now, Player B face imperfect information. However, if \(c_{B1}, c_{B2}, c_{B4} \succ c_{B3}, c_{B5}\) then it decides to move up (\(\uparrow \)) independently whether it is at \(B_1\) of \(B_2\). Although it does not know how much it gets, it, at least, knows that moving up is the optimal action. To indicate this partial knowledge we call the boundary weak.

Strong Boundaries. Like before, we assume \(c_{A6}, c_{A7} \prec c_{A1}, c_{A2}, c_{A3}, c_{A4}, c_{A5}\), i.e., Player A will go up (\(\uparrow \)) or to the right (\(\rightarrow \)). In the case of strong boundaries, Player B does not know what it will get but it also does not know its optimal action at the information set \(\{B_2, B_3\}\). E.g., for \(c_{B1}, c_{B2}, c_{B5} \succ c_{B3}, c_{B4}\) Player B should move up (\(\uparrow \)) if it is at node \(B_2\) but should move down (\(\downarrow \)) if it is at \(B_3\). To indicate this absence of any knowledge we call the boundary strong (dominating).

## 3 Incomplete Information

### 3.1 Incomplete Information in Extensive Form Games

Player B moves up (\(\uparrow \)) \(\Rightarrow \) Player A moves also up (\(\uparrow \)) since \((c_{A1} = 10) \succ (c_{A2} = 5)\). Player B obtains \(c_{B1} = 8\).

Player B moves down (\(\downarrow \)) and obtains \(c_{B3} = 9\).

Obviously, in the case of complete information, Player B would decide to move down (\(\downarrow \)).

Now, we assume that Player B faces incomplete information, i.e., it does not know the possible payoffs of Player A (as in the right sub-figure of Fig. 2). Therefore, it does not know if Player A will move up or down if it is at node \(A_2\).

### 3.2 Incomplete Information and Rough Sets

To discuss incomplete information in extensive form games we refrain from the possible transformation, the so called Harsanyi transformation [5], of a game with incomplete information into a game with imperfect information. A rough interpretation would go beyond the scope of our paper.

Incomplete information: rough strategies of Player B

Rough strategy | Player B | Player A | Payoffs | |
---|---|---|---|---|

Action | at node | Action at \(A_2\) | ||

\(b_1\) | \(\uparrow \) | \(B_1\) | \(\uparrow \) | \(C_1\) |

\(b_2\) | \(\uparrow \) | \(B_1\) | \(\rightarrow \) | \(C_2\) |

\(b_3\) | \(\downarrow \) | \(B_1\) | \(*\) | \(C_3\) |

## 4 A Rough Payoff Matrix

Imperfect and incomplete information: rough strategies of Player B

Rough strategy | Player B | Player A | Payoffs | |
---|---|---|---|---|

Action | at node | Action at \(A_1, A_2\) | ||

\(b_1\) | \(\uparrow \) | \(B_1\) | \(\uparrow \uparrow \) | \(C_1\) |

\(b_2\) | \(\uparrow \) | \(B_1\) | \(\uparrow \downarrow \) | \(C_2\) |

\(b_3\) | \(\uparrow \) | \(B_2\) | \(\rightarrow *\) | \(C_4\) |

\(b_4\) | \(\downarrow \) | \(B_1\) | \(\uparrow *\) | \(C_3\) |

\(b_5\) | \(\downarrow \) | \(B_2\) | \(\rightarrow *\) | \(C_5\) |

\(b_6\) | \(\uparrow \) | \(B_3\) | \(\downarrow *\) | \(C_6\) |

\(b_7\) | \(\downarrow \) | \(B_3\) | \(\downarrow *\) | \(C_7\) |

Boundary. We define a boundary when a strategy of a player can lead to more than one payoff.

Lower Approximation. Any strategy that is not a member of a boundary belongs to a lower approximation, i.e., a strategy of a player leads to one and only one possible payoff.

*R*(with a hat (\(\widehat{R}\)) indicating a boundary and an underline (\(\underline{R}\)) a lower approximation):

Rough payoff matrix

The rough payoff matrix discloses structures of the corresponding extensive form game regarding imperfect and incomplete information faced by Player B. The payoffs \(\widehat{c}_{B1}, \widehat{c}_{B2}\) and \(\widehat{c}_{B4}\) form the boundary region \(\widehat{R}_1\) and the payoffs \(\widehat{c}_{B2}\) and \(\widehat{c}_{B4}\) the boundary region \(\widehat{R}_2\), while the payoffs \(\underline{c}_{B6}\) and \(\underline{c}_{B7}\) belong to separate lower approximations: \(\underline{R}_3\) and \(\underline{R}_4\), respectively.

The payoffs can be characterized with respect to their degree of ‘roughness’ \(\rho \), the percentage of boundary payoffs of the players. For \( C_1 = (\underline{c}_{A1}, \widehat{c}_{B1})\) we would get \(\rho (C_1) = 1/2 =0.5\) indicating that one out of two payoffs belong to boundaries; similarly, e.g., \(\rho (C_7) = 0/2 =0.0\). It also makes sense to distinguish payoffs derived from imperfect and incomplete information. Some implications of the selection of the (rough) equilibria have already been discussed in the previous sections. They can be determined straightforwardly from Table 6. Therefore, we refrain from a detailed discussion here.

## 5 Conclusion

In this paper, the relationship between imperfect information and incomplete information in game theory and rough sets is discussed. While in both areas great attention has been given how to deal with information a discussion of their relationship is still missing. We showed how imperfect information and incomplete information in game theory can be interpreted in rough set terms. The information sets comprising of two or more nodes in extensive form games are similar to boundaries in rough sets and, therefore, can be interpreted from a rough set perspective. We limited our examples to illustrative and simple cases to motivate for further research in this area. A more detailed discussion on the relationship of imperfect and incomplete information in classic game theory and rough sets could be mutually beneficial to both fields. It possibly leads to applications beyond game theoretic rough sets; in particular, it would be interesting to investigate the potentials of an integrated ‘rough game theory’.

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