# Formal definition of perfect information

I am looking for the formal definition of 'perfect information' in game theory.

Please direct me to a book or preferably an online paper where I can find it.

On a related note:
The Wikipedia page for the term is not very useful. It only offers an informal definition:

In game theory, an extensive-form game has perfect information if each player, when making any decision, is perfectly informed of all the events that have previously occurred

Given this definition the simultaneous move examples seem strange. The games mentioned (e.g. iterated prisoner's dilemma) could easily be altered to have sequential moves where the second mover is simply not aware of the first move. This game would have the same extensive form but would no longer fit the informal definition.

# Osborne and Rubinstein (1994)

Osborne and Rubinstein's textbook A Course in Game Theory defines (extensive) games with perfect information in three versions.

• Basic version (Def. 89.1): perfect information is the same as requiring information sets be singletons (though they don't put it this way). In the author's language, perfect information is modeled as a player function $P:H\to N$, mapping each non-terminal history $h\in H$ of previous moves to a single member in the set of players $N$. A history here is $h=(a^k)_{k=1,\dots,K}$, where $a^k$ is the action taken by the player who moves in the $k$th round, and $K$ is possibly infinite.

• Extended version 1 (perfect information with chance-moves, Sect. 6.3.1): perfect information (for players) here is basically the same as before, be the definition incorporates uncertainty in the game due to chances

• Extended version 2 (perfect information and simultaneous moves, Sect. 6.3.2): perfect information here is modeled as a player function $P$ mapping each non-terminal history to a set of players, where

[a] history in such a game is a sequence of vectors; the components of each vector $a^k$ are the actions taken by the players whose turn it is to move after the history $(a^\ell)_{\ell=1}^{k-1}$. The set of actions among which each player $i \in P(h)$ can choose after the history $h$ is $A_i(h)$; the interpretation is that the choices of the players in $P(h)$ are made simultaneously.

# Myerson (1991)

Myerson's Game Theory: Analysis of Conflict similarly defines (on page 185) an extensive form game with perfect information as information sets being singletons within each information state.

# Fudenberg and Tirole (1991)

Fudenberg and Tirole's Game Theory textbook defines perfect information informally (on page 72) as follows

We say that multi-stage game has perfect information if, for every stage $k$ and history $h^k$, exactly one player has a nontrivial choice set --- a choice set with more than one element --- and all the others have the one-element choice set "do nothing."

A simultaneous-move game is not a game of perfect information. It is a game of imperfect information.

Let me quote Gibbons (Chapter 2, p.58).

The key features of a dynamic game of complete and perfect information are that (i) the moves occur in sequence, (ii) all previous moves are observed before the next move is chosen, and (iii) the players' payoffs from each feasible combination of moves are common knowledge.

Picking up the discussion in the comments, let me again quote Gibbons, p.122, Footnote 19:

This characterization of perfect and imperfect information in terms of singleton and nonsingleton information sets is restricted to games of complete information because, as we will see in Chapter 4, the extensive-form representation of a game with perfect but incomplete information has a nonsingleton information set. In this chapter, however, we restrict attention to complete information.

• The problem with this is that it also says "dynamic game" where some moves will occur in sequence. Thus the sequential move part may not a property implied by perfect information. According to this the iterated PD example on Wikipedia is incorrect though. – Giskard Feb 10 '17 at 15:31
• Seems to me that if I were to take away the dynamic part I would basically be left with 'perfect information means singleton information sets'. Is that right? – Giskard Feb 10 '17 at 15:37
• Yes, perfect information means all info sets are singletons. – Bayesian Feb 10 '17 at 15:38
• en.wikipedia.org/wiki/Extensive-form_game (CTRL+F "singleton") - but the examples in the other wiki article you shared are indeed confusing. "Information sets are singletons" is the version I teach. I check for a "higher authority" later today. – Bayesian Feb 10 '17 at 15:44
• I corrected my statement, sorry for that. I thought about complete info as well. – Bayesian Feb 10 '17 at 16:03