Games with Unawareness

1.1 Three Examples

We operationalize restricted perception of the game as follows: A player – Alice – can be unaware of some of the actions available to another player – Bob. This will imply that the game she perceives does not contain some of Bob’s actions. Alternatively, Alice may not include a third player in the game at all. This again represents a restricted game. It might also be the case that Bob perceives that Alice is unaware of some aspect of the game, in this case his view of her view of the game is restricted but it does not imply that Alice is indeed restricted in this manner. It might well be that Alice is aware of all aspects of the interaction, yet Bob’s perception of her perception is limited. The two examples below provide some variations on these type of situations. They also shed some light on how the solutions to these games are constructed. The guiding principle for behavior is that every player chooses a strategy in the game they perceive which reacts (e.g., with a best response) to strategies the player believes others will be playing in the game she thinks they perceive.

Our first example is of a normal form game with unawareness. We begin with the game depicted in (1) below. This game represents all the actions available to the players – Alice and Bob – and the payoffs associated with each action profile. Assume that in the situation we are modeling Alice and Bob are both aware of all the actions available in the game, so when they write a description of the actions and payoffs it corresponds to (1) below. However, Alice is unaware that Bob is aware of all her actions. In particular, She thinks that he is only aware of the actions { a 1 ,   a 2 ,   b 1 ,   b 2 ,   b 3 } – She does not realize that her third action is included in the game as Bob perceives it. This situation may arise if Alice perceives that action a 3 is secret, or if, say, Bob is new to the environment in which this interaction occurs and there is no reason to think he would model action a 3 .

This situation may arise if Bob actually learns about Alice’s secret action, or talks to an “old-timer” who reveals the existence of the not so obvious third action available to Alice. We note, that the strategy profile ( a 2 ,   b 1 ) is the unique NE of the game depicted in (1), hence if such games have been played in the past (even if fully perceived by all players) the action a 3 may not have been observed. Hence, even if Alice believes an inexperienced Bob will study similar past situations, she may well conclude that he will not model this action. In this example we further assume that Bob realizes all this. Not only does he consider a 3 he also deduces that Alice does not realize that he is considering it, e.g., if he learned it was a secret action. Hence, Bob is aware that Alice perceives Bob’s perception of the game to consist only of the actions { a 1 ,   a 2 ,   b 1 ,   b 2 ,   b 3 } . We also assume that Bob is aware that Alice is aware of the whole action set { a 1 ,   a 2 ,   a 3 ,   b 1 ,   b 2 ,   b 3 } .

Turning to higher levels of interactive views of the game; since Alice is only aware of Bob being aware of { a 1 ,   a 2 ,   b 1 ,   b 2 ,   b 3 } , we assume she cannot be aware that he is aware that she is aware of anything beyond this set, otherwise, she would be aware that he is able to reason about her reasoning about the additional action a 3 , so he must be able to reason about a 3 as far as Alice can deduce, which would contradict our assumption that Alice is unaware that Bob is aware of a 3 . In particular, any higher order iteration of awareness of Alice and Bob which is not considered above is assumed to be associated with the set { a 1 ,   a 2 ,   b 1 ,   b 2 ,   b 3 } .

As noted above, the initial game we started out with in (1) has a unique NE ( a 2 ,   b 1 ) obtained by iteratively eliminating strictly dominated strategies. However, while both players are aware that this is the game being played, we assumed that Alice perceives that Bob is aware of only two of her actions. In other words, Alice is unaware that Bob is aware of a 3 . While Alice and Bob both view the game as in (1) Alice perceives that Bob finds the game being played as depicted in (2).

Hence, Alice also finds that Bob finds that she perceives the game as in (2), and so on for every higher order awareness. We obtain that Alice perceives that Bob views the game as a standard normal form game with complete awareness – a game where all participating players are aware of all aspects of the game, they are aware that all other players are aware of the same game, and so on. Taking Nash equilibria as the solution concept for normal form games, Alice may deduce that Bob plays according to the NE ( a 1 ,   b 2 ) of the game in (2) which is also the Pareto dominant outcome of this normal form game. Alice, who sees herself as being more aware than Bob, will be inclined to choose her best response to b 2 which is a 3 in the game as she perceives it. Bob can make the exact same deduction that we, as modelers, just made, since he is aware of all the actions and he fully realizes how Alice perceives his awareness. Hence, Bob may deduce that Alice, being unaware of his full awareness, will assume he plays b 2 and she will play a 3 as her best response. Bob being aware of all that can deduce this behavior based on the iterated best response principle. This would lead Bob to play his best response to a 3 which is b 3 . We will have that Alice chooses a 3 and Bob chooses b 3 as a result of this higher order unawareness. Starting with a NE and considering best responses, we end up with the worst possible payoff for Alice and a payoff below all Nash equilibria of either games for both players, even though both are aware of the full extent of the game, both are commonly aware of the action profile of the unique NE ( a 2 ,   b 1 ) in that game and both act rationally given their perceived view of the game and assume the other player does as well.

This example illustrates the general formulation of a game with unawareness as a collection of standard normal form games: A game describing how each player views the game, a game describing how each player views another player’s view of the game, games describing how they view how others view others’ view of the game and so on. We also introduced one of the properties linking these viewpoints when we assumed that if Alice is aware that Bob is aware of X she is also aware of X. This corresponds to the interpretation that if Alice thinks Bob thinks X is an important part of the game, she can definitely reason about X and would model and consider it as well. We also introduced how solutions can be extended by assuming players use reasoning principles based on solutions in the games as they perceive them while taking into account other players will apply those principles in the games they perceive. Here we have Alice, in the game as she perceives it, plays a best response to a strategy of Bob in the game Alice think that Bob is considering and Bob using this reasoning w.r.t. the viewpoints he perceives to devise his best response.

Our second example applies the same principles of modeling and reasoning to a repeated game setting. In this example a simple twist in the players unawareness leads to a cooperative result based on best response reasoning. Consider the following prisoner’s dilemma repeated, say ten times:

We denote this stage game by G 1 . Assume both Alice and Bob are fully aware and perceive the game being played is G 1 . However, both players initially perceive that the other player is unaware of the non-cooperative action, i.e., each player perceives the other player as perceiving the game as follows:

Let us denote the game with the single cooperative action by G 2 . As the game is played repeatedly these perceptions remain unchanged as long as the players both cooperate, however if any one of the players defects the players immediately become commonly aware of the full game, in other words they attribute the same full perception of the game as G 1 to each other iteratively to any high order.

We can capture this situation with the following representation. For a given history of play h we can denote by G ( h , i ) how player i views the game after observing the history h. Similarly, we denote by G h , i ˆ h ’ , j how i after history h perceives that j ≠ i will perceive the game after j observed history h ’ . In our example, G ( ϕ , i ) = G 1 , G ϕ , i ˆ ϕ , j = G 2 , G ϕ , i ˆ ϕ , j ˆ ϕ , i = G 2 and so on. Similarly, as long as the histories h ,   h ’ ,   h ’’ being considered only include the cooperative actions ( a 1 ,   b 1 ) we have G ( h , i ) = G 1 , G ( h , i ) ˆ ( h ’ , j ) = G 2 , G ( h , i ) ˆ ( h ’ , j ) ˆ ( h ’’ , i ) = G 2 and so on. However, if defection occurs the perceptions change. Hence if either ( a 2 ,   b 1 ) , ( a 1 ,   b 2 ) or ( a 2 ,   b 2 ) has occurred in h ’ we have that G ( h , i )   ˆ ( h ’ , j ) = G 1 . In particular, player i realizes that if defection is played then player j’s perception will include the complete game.

The description above does not paint a full picture of how the players perception of the game might evolve. For example, after seeing defection by Alice it could be the case that Bob realizes that Alice was aware of the full game all along, or Bob may reason that she just recently became aware of the possibility of defection. These are different scenarios and can be captured by correspondingly completing the definition of various higher order perceptions in this game. For example, we can assume that G ( h , i )   ˆ ( h ’ , j )   ˆ ( h ’’ , i ) = G 1 when defection has occurred in h ’ for all h ’’ including say h ’’ = ϕ . This amounts to assuming that i believes that once defection occurs then j will realize that i was aware of the full game all along. Or, we can model a situation were i thinks that j thinks that i learned about the defection action, rather than perceived it all along. Similarly, further variations could represent a difference in how higher order perceptions evolve based on the particular player who defected. For example, if in the history h ’ player i was the first to defect then player j may reason that player i knew about defection all along, while if it was only j that defected first, i may think that j only learned about defection at that point in time from her. Clearly this can be further complicated as one considerers various higher level bifurcations of how players interpret each others’ past perceptions based on learning something unexpected.

No matter how we complete some these higher order perceptions, it is important to note that the only perceptions relevant are those that the players think possible. Hence, since G ( ϕ , 1 )   ˆ ( ϕ , 2 ) = G 2 we do not consider the perception of ( ϕ ,   1 ) – Alice before actions are taken – of the perception of ( ( a 1 ,   b 2 ) ,   2 ) , since Alice at the beginning does not think that Bob is reasoning about action b 2 . In other words we do not consider the following iterated view of the game G ( ϕ , 1 )   ˆ ( ( a 1 , b 2 ) , 2 ) as relevant. Hence, the higher order perceptions that are considered respect the restrictions of reasoning that the players attribute to one another for each perception associated with a player and a relevant history of play.

We can now turn to a quick intuitive analysis of reasoning in this example. Each player reasons that if they choose to defect at any point it will cause the common perception that the game being played is G 1 . Hence, each player would anticipate that as long as they play cooperatively, they will receive a payoff of 2 since they assume the other player will play cooperatively as they perceive them to be unaware of any other action. Each player will reason that playing cooperatively until the final round and defecting in round ten would lead to a payoff of 2 * 9 + 3 = 21 from their perspective. Any deviation earlier would lead to a payoff of 3 instead of 2 in that early round of assumed unilateral defection, followed by defection by both players until the end if we assume any of the standard solution concepts to the remaining commonly perceived finite prisoner’s dilemma. Hence the maximum payoff attained by defecting earlier is strictly lower. We have that both players will therefor play cooperatively in all nine rounds and defect in the last one. Needless to say that both players would be surprised to see the other player defect in the last round. The best response reasoning we described is based on assuming that once the game unravels to be G 1 both players will deviate throughout which is supported in the extension of standard solution concepts to our game forms.

We note that one can consider higher order variations of this example, such as a case where each player perceives the full game G 1 and perceives that the other player also perceives the game as G 1 but that the other player perceives them as perceiving the game as G 2 , in other words G ( ϕ , i )   ˆ ( ϕ , j ) = G 1 and G ( ϕ , i )   ˆ ( ϕ , j )   ˆ ( ϕ , i ) = G 2 . In this case, under the assumption that any defection unravels any misperception and that reasoning in the unraveled game leads to defection throughout, there is a cooperative solution where the players will cooperate up to round eight, and then defect in the last two rounds expecting the other player to defect only in round ten.

Our third example applies the same principles of modeling and reasoning to a dynamic setting which allow us to also capture how the players’ perceptions might change or be changed strategically. We begin with a story in which Alice – the baker – has contracted with Bob – the coffee shop owner – to produce and deliver baked goods from her bakery to his coffee shop. Alice and Bob have contracted on the baked goods based on their expected costs and revenues from the transaction. Alice’s cost for producing and delivering the baked goods on Monday is 4. The value to Bob if the goods arrive on Monday is 7 and we assume they contracted on a price of 5 for the baking and delivery of the goods on Monday. Hence, their payoffs will be 1 ,   2 respectively. However, Alice can also bake the goods and deliver them on Tuesday. Given how busy Monday is, her cost of baking and delivering the goods on Tuesday is lower and stands at 2, the value to Bob for receiving the goods on Tuesday is much lower and stands at 3 since many of his customers prefer to purchase the goods on Monday. Anticipating that, Alice and Bob have stipulated in the contract that if Alice delivers the goods on Tuesday Bob will only need to pay 2 instead of 5. Leading to a final payoff 0,1 to Alice and Bob respectively. This situation so far is depicted in Figure 1.

Figure 1:

Alice’s shipping decision.

The contract also stipulates that in the case of unforeseen contingencies beyond her control that prevent Alice from delivering the baked goods on Monday, she is expected to deliver them on Tuesday but will be paid the full 5 payment at that point and not the reduced payment. The payoffs may depend on the particular contingency. For example, if Alice produces the goods but her delivery truck meets an unforseen contingency she may need to produce and deliver the baked goods again on Tuesday which will cost her a total cost of 6 = 4 + 2 . In such a case, even though she is paid 5, she ends up with − 1 . While, if there is a severe snow storm on Monday and roads are closed, she could reasonably decide not to produce on Monday and produce and deliver on Tuesday and incur only a cost of 2 and receive 5 from Bob to end up with 3.

Driving to his coffee shop early Monday morning Bob notices that unanticipated major road work is scheduled for that day (and that day only). He estimates that if Alice attempts to deliver the baked goods on Monday then there is a 50 % chance it will not arrive on time and in that case the baked goods will be ruined. Bob is confident that Alice is not aware of the pending construction since the bakery is in the other side of town and he knows Alice lives close to the bakery. Bob realizes that the strategic situation is no longer as he envisioned it and was depicted in Figure 1. Bob realizes that there was a nature move that resulted in major road delays. Moreover, Bob realizes that he now has new possible actions; he can either call Alice and let her know about the road work or not.

Let’s consider the payoffs given this new situation. If Bob does not call Alice then Alice will not know about the possibility of road work, she will be unaware of it and assume the game is as depicted in Figure 1. However, the payoffs will be different. If Alice attempts the delivery on Monday then with 50 % chance it will go through and Bob’s payoff will be 2, at the same time Alice will receive 1 if it gets through on time. With 50 % chance the delivery will not go through and in this case Alice will need to bake and deliver the goods again on Tuesday. In that case unforeseen contingencies will be invoked and Alice will still receive the full payment on Tuesday. Bob’s payoff will be − 2 = 3 − 5 , at the same time Alice will receive − 1 = 5 − 6 since her cost will include the production and delivery costs for both days. Note that these are the eventual payoffs and are also the payoffs as perceived by Bob’s viewpoint. But when Alice makes her decision wether to make and deliver the goods on Monday or Tuesday, she is unaware of these payoffs and believes the game is as depicted in Figure 1.

If Bob decides to tell Alice about the impending road work, the expected payoffs for an attempted delivery on Monday will not change for either player. But now Alice will be aware of the unforeseen contingency and realize that road work is possible and actually occurred. At that decision point she can invoke it as a justifiable reason to produce and deliver the baked goods on Tuesday without penalty based on the unforeseen contingency clause. Hence, her payoff will be 3 = 5 − 2 , while Bob will have to pay the full price and end up with − 2 = 3 − 5 .

We depict this game with unawareness using extensive form games as they are perceived by the players in every decision point. The game with unawareness we just described is depicted in Figure 2. The game includes Nature choosing whether there is road work. If there was no road work we have the game depicted in blue which represents how Alice (and Bob) perceive the game when she is unaware of the road work. If road work occurs as in our story (or considered explicitly), Bob can choose whether to tell Alice or not. If he doesn’t tell Alice she is still unaware of the road work although it happened, hence she perceives herself to be at a different decision point corresponding to the game depicted in Figure 1 imbedded in the game depicted in Figure 2 after the “no road work” option. We depict this link as a blue arrow indicating where Alice perceives the game is. Hence the game depicted in blue corresponds to the game when there is no awareness of the road work or its possibility. If Bob does let Alice know about the road work she becomes aware of it and realizes the full scope on the game and takes an action at that decision point. The game as perceived by Bob after he learns of the road work and will be perceived by Alice if he tells her is the combined game in black and blue. Whereas aware Bob and Alice also view Alice’s view of the game when she is unaware as denoted by the blue game and the blue arrow.

Figure 2:

Bob’s Calling Decision. This is the game as seen by Bob, if he does not call Alice he thinks that Alice views the game as in Figure 1 depicted in blue (the arrow indicates that in both decision points of the information set the Alice perceives the same game, in particular she is unaware that there is an information set).

Given the description of the game we can now consider the players’ behavior. Using a backward induction principle it seems that Alice would choose to produce and deliver on Monday if she believes the game is as depicted in Figure 1. Hence, she is expected to do so also if there is road work but Bob does not call in the game in Figure 2. If Bob does call she is likely to choose Tuesday since she becomes aware of the road work. Hence, Bob can conclude that his payoff will be − 2 if he calls Alice and 0 if he doesn’t, hence when he observes the road work he will, by backward induction, not call Alice leading to a payoff of 0,0 . Hence, we applied a backward induction reasoning where each player is considering how other players in the future will apply backward induction in the game which the player thinks they will perceive.

Consider now another twist of the baked goods delivery scenario. This time, it turns out that Carol who works for Alice at the bakery lives nearby the road construction area and observes the unanticipated road work as well. Carol has two actions, she can text Alice about the potential road work or not. Her payoff for reporting it is 1 while her payoff if she does not report it is 0. Had there been no road work Carol’s payoff is assumed to be 0 as well. In this case Bob is unaware of Carol’s actions, in fact he may be unaware of Carol’s existence and may have not modeled the possibility of an employee living next to the road work. Hence, Bob’s view of the game as is we described in Figure 2. However, the game now has a third player, Carol and her action can impact the awareness of Alice. We depict the game in Figure 3. Here we have the game as originally modeled by Alice in blue, the games in green and blue describe how Bob views the game and how he views Alice will view the game if he does not call her to let her know about the road work (in blue) as well as if he does call her (in green). The game now also has Carol’s decision points and payoffs (in black) as well as how will Alice perceive the game if Carol texts her about the road work (in black). Hence, if Carol texts Alice then Alice will view the game as in Figure 3, while viewing Bob’s view of the game if he observes road work as depicted in the game with unawareness in Figure 2 which includes how Bob views Alice’s view of the game when she is unaware of road work as in Figure 1. The payoffs remain unchanged from the previous game and we just note that Alice’s payoff when Carol texts her is now 3 (and Bob’s − 2 ) if she produces and delivers on Tuesday no matter if Bob calls her or not, since she can now invoke the unforeseen contingency. In fact, we can imagine a situation where her payoff is even higher if Bob does not call her especially since she is aware that he is aware of the road work^[1].

Figure 3:

With Carol’s E-mail Decision.

This is the game as seen by the modeler as well as by Alice once Carol texts her. The game in green and blue indicates the game as Bob is aware of once he learns about the road work, it is also how Alice views Bob’s view of the game once she knows about the road work. The game in blue is the game that Alice plays if she is unaware of the road work, what Bob thinks she plays if he doesn’t call her and what Alice thinks Bob thinks she thinks the game is if he does not call her.

The game form illustrates the nested nature of higher order view points. In this game the green decision points and actions constitute a subtree hence naturally correspond to a game tree. Furthermore, there is unawareness of a player (Bob is unaware of Carol). The reasoning can be applied similarly as before, Carol will text Alice who will then become aware of the full scope of the game with unawareness and will choose to invoke the contingency and deliver on Tuesday. Bob will not call Alice being unaware of Carol and he would expect Alice to try and ship on Monday.

The extensive form game with unawareness is a collection of standard extensive form games corresponding to how Alice perceives the game at her decision points, how Alice perceives how Bob perceives the game at his decision points in the game she considers, how she perceives the game he perceives that she perceives in her decision point in the game she perceives he is considering, and so on. These nested games will have consistency conditions that assure principles such as Alice being aware of what she thinks Bob is aware of, as well as making sure that the view from each decision point is indeed itself a game with unawareness. As with the extension of normal form games, solution concepts can be extended by applying reasoning principles to the decision points taking into account that these will be applied to the nested extensive form games as they are perceived in those decision points.

1.2 Results and Related Literature

The examples above are meant to illustrate a number of features of our model. First, the definition of games with unawareness for all game forms is based on a collection of standard games, games that describe players’ viewpoints, views of other players views of the game and so on. In the case of extensive form games, viewpoints correspond to decision points, in incomplete information games defined below they will correspond to types. We can also see the nested features of these games where the actions and players that Alice views Bob is considering become part of the game that Alice is considering. Hence the game form satisfies the property that if Alice thinks Bob is aware of an aspect of the game she is also aware of that aspect. The examples also demonstrate how solutions can come about in these games as they illustrate the best response reasoning as Alice considers the actions of Bob in the game she perceives he is considering.

After defining the extended forms for normal, repeated, incomplete information and extensive form games with unawareness, our structural result, Proposition 1, shows that all the definitions satisfy the property that each player’s view of the game, view of others, and so on, itself is a game with unawareness satisfying the exact same conditions as we, the modelers, are considering. In particular, each view of each other player induces a game with unawareness and so on at higher levels of iterative views. Hence we have a common perception that all players are modeling a game with the same framework. Next we extend rationalizability and Nash equilibria to normal form games with unawareness and prove existence in these possible infinite structures using the generalizations of equilibria existence conditions due to Glicksberg (1952) and Fan (1952). We similarly define extended BNE for incomplete information games with unawareness and extend the notion of assessment to extensive form games with unawareness allowing the definition of assessment based equilibria (such as sequential equilibria). We then demonstrate that all extended solution concepts are non-empty (assuming standard conditions for the underlying perceived games) and that the extended solutions coincide with the standard solutions when there is no unawareness. Games with unawareness may have infinite hierarchies of the perceptions of the game yielding an infinite number of different viewpoints within a single game with unawareness. Using a result by Higman (1952) we provide sufficient conditions for a game with unawareness to induce only a finite collection of viewpoints and hence have a finite representation.

One of the main questions that arises with any new game form is whether it can tell us something that the standard formulations cannot. With restricted perceptions it is particularly important to study whether these situations can be modeled using probability zero rather than unawareness. We show that indeed normal form games with unawareness can be represented as games with incomplete information. Moreover, there is a natural mapping (in the sense that the mapping is independent of the exact payoffs in the game with unawareness) to incomplete information games that fully captures all the relevant views in the game. But while the structure can be mapped between game forms the extended NE in games with unawareness does not map to any known solution of Bayesian games. In fact, it seems that generating such a solution in the Bayesian game equivalent would essentially amount to reconstructing the normal form game with unawareness. We show that the extended NE is a strict refinement of the Bayesian NE of the incomplete information representation, while, interestingly, the latter coincides with extended rationalizability for normal form games with unawareness.

There are two general types of studies that relates to multi-person unawareness in economic theory. The first is a logic and foundational perspective. One of the earliest works providing foundations for unawareness is Fagin and Halpern (1988) who provide a host of logics for unawareness and study their syntax and semantics. They mention the “semantic naturalness” of their modeling as it provides a clear interpretation of the notion of awareness. Halpern (2001) and Halpern and Rego (2008, 2009, 2013) expand the analysis as well as provide more insights on the properties of unawareness, with the latter two papers providing important insights on the foundational question of modeling of awareness of unawareness. Modica and Rustichini (1994, 1999) have developed a decision theoretic treatment of unawareness based on the notion of event-based reasoning (partition models). The work of Heifetz, Meier and Schipper (2006, 2008) took an event-based approach to interactive unawareness. This semantic approach provided a natural framework for interactive unawareness that also allows to bypass the impossibility results of the classical partition model framework. Li (2008a, 2009) provided an alternative product based construction for modeling unawareness and Sadzik (2006) developed a novel framework for the foundations of probability and unawareness. Board and Chung (2007) propose an alternative approach and an axiomatization that also touches on awareness of unawareness and Walker (2014) provides an alternative semantic approach.

The second type of studies considers the modeling of games with unawareness, the extension of existing solution concepts and exploration of new ones as well as applications to particular economic settings. Many of the foundational papers were expanded or include the development of tools for modeling strategic interaction with unawareness. Copic and Galeotti (2005) consider awareness action equilibrium in the context of interactive awareness. Li (2006) considers unawareness in dynamic games. Mengel, Tsakas and Vostroknukov (2009) model and study unawareness in repeated games. Grant and Quiggin (2013) provide an alternative model for extensive form game and Halpern and Rego (2014) extend their framework to extensive form game and define Nash equilibria while Rego and Halpern (2012) define sequential equilibria. Meier and Schipper (2014) provide a model for Bayesian games with unawareness, Heifetz, Meier and Schipper (2013a) apply their foundational approach to strategic introductions and specifically explore characteristics of the no-trade theorem. In Feinberg (2004a) a syntax based approach to interactive unawareness is introduced and applied to extensive form games. Heifetz, Meier and Schipper (2013b) consider the application to dynamic games and extend extensive form correlated rationalizability to these games. Board and Chung (2009) explore the application of their axiomatic model to interactive unawareness and Sasaki (2014) defines subjective rationality in games where players may misspecify the game. Additional studies both model and apply unawareness in interactive settings to particular classes of economic problems. Chen and Zhao (2013) explore unawareness in principle-agent models and von Thadden and Zhao (2012) study incentive schemes in the presence of unawareness and in von Thadden and Zhao (2013) explore multi dimensional principal agent problems with unawareness, Zhao (2011) studies the framing of contingencies in contracts, Filiz-Ozbay (2012) studies unawareness modeling in contract theory, Grant, Kline and Quiggin (2012) explore different notions of awareness and ambiguity in contracts and Zhao (2008) studies moral hazard with unawareness. Our approach extends the conceptual principle of modeling the subjective reasoning of players from the perspective of every decision instance which we developed in Feinberg (2005b, 2005c). Finally, the work in Feinberg (2005a) contains initial results regarding modeling normal form games with unawareness and is subsumed by the current paper. For more related work on unawareness also consult the bibliography set up by Burkhard Schipper at https://www.econ.ucdavis.edu/faculty/schipper/unaw.htm.

The use of standard games as the primitive building block for games with unawareness allows us to keep the definitions relatively short and simple with four conditions applied to each game form and reduces the variation between the game form being used. The nested structure of standard games also provides a relatively clear extension of existing solution concepts to games with unawareness. Using hierarchies of perceptions allows the proof of consistency of the representation in the sense that the modeler, players, and higher order players’ view of other players, all have the same form as games with unawareness. The importance of this characteristic is evident if we view our modeling as prescriptive for reasoning agents and, therefore, want its descriptive nature to coincide with how the reasoning agents are assumed to model their strategic environment.

2.1 Normal Form Games with Unawareness

In standard normal form games the modeler describes the set of players, their possible actions and payoffs for action profiles. The modeler’s normal form game is our starting point: G = ( I ,   ∏ i ∈ I A i ,   { u i } i ∈ I ) where I is a finite set of players, A i is the finite set of actions available for each player and the functions u i associate the utility for action profiles in ∏ j ∈ I A j . These are the set of players and actions that the modeler is considering whether or not the players are aware of each other or of some of the actions. Each player may have a restricted view of the game. Hence for a player v ∈ I we consider a normal form game G v = ( I v ,   ∏ i ∈ I v ( A i ) v ,   { ( u i ) v } i ∈ I v ) . Similarly, a player considers how each of the players that appear in her game models the game. In general, a finite sequence of players υ = i 1 ,   … ,   i n is associated with a normal form game G υ = I υ ,   ∏ i ∈ I υ A i υ ,   u i υ i ∈ I υ where I υ is the set of players that i 1 finds that i 2 finds that … that i n is considering, and similarly for the sets of actions A i υ and payoffs u i υ defined on the set of action profiles ∏ i ∈ I υ A i υ . We call υ an iterated view, or in short a view. Throughout, υ = ϕ corresponds to the modeler’s view, i.e., G ϕ = G . Note that this is the modeler’s view of the relevant players and their actions and payoffs not of the unawareness of players. Correspondingly, we say that G υ is the situation as viewed, or perceived at υ, that A j υ is the set of j’s actions as viewed from υ and so on. We denote an action profile in G υ by a υ . The singletons ( v = i ) corresponding to a player’s view are called viewpoints and the set of viewpoints (players, in the case of normal form games) is denoted V with a typical element v.

We denote the concatenation of two views υ ‾ = i 1 ,   … ,   i n followed by υ ˜ = j 1 ,   … ,   j m as υ = υ ‾ ˆ υ ˜ = i 1 ,   … ,   i n ,   j 1 ,   … ,   j m . The set of all potential views is denoted V ‾ = ∪ n = 0 ∞ ( I ) ( n ) where I ( n ) = ∏ ​ j = 1 n   I and I ( 0 ) = ϕ .

Definition 1. A collection Γ = G υ υ ∈ V where G υ are normal form games and V ⊂ V ‾ is a collection of finite sequences of players is called a normal form game with unawareness and the collection of views V is called its set of relevant views if the following properties hold:

C1 For every υ ∈ V we have

The first condition requires that the set of relevant views V is closed under the set of players considered in the game perceived at a relevant view and that viewpoints of non-players are irrelevant. There would be no impact on our results if players were to consider the views of players not participating in the game, however we find such redundancy unpleasing, as with any scientific modeling. The other direction of this condition is crucial for our setting since if Alice models Bob as one of the players in the game, it is required that Alice should find Bob’s view of the game to be relevant.

C2 For every υ ˆ υ ˜ ∈ V we have

The first part of this condition states that if a relevant view’s perception of another view is relevant, then the first view must itself be relevant, e.g., if it is relevant to consider Alice’s view of Bob’s view of Carol, then Alice’s view of Bob is also relevant. Together with condition C1 this implies that the set of relevant views is exactly the set of views inductively constructed from considering the players that are perceived to be participating in the game. In particular, if there exists any relevant view at all we also have, fortunately, that ϕ ∈ V – the modeler’s view is relevant.

Condition C2 extends this principle to the set of players and actions: if Alice finds that Bob is considering a player or an action as part of the game, she herself must consider them to be part of the game. In other words, by the mere fact that you find it relevant that others find some aspect of the game to be relevant enough for modeling, you must model that aspect yourself. Much like we, the modelers, do when considering the reasoning of players. Restating this with the notation of awareness the condition states that: what Alice is aware that Bob is aware of, are things that Alice is aware of as well.

C3 If υ ˆ v   ˆ   υ ‾ ∈ V we have

The third condition requires that each player, that is relevant along some view, has a correct perception of their own perceived perception: if Alice perceives that Bob has a certain perception of the game, she also perceives that he perceives to have that perception. With awareness: if Alice is aware that Bob is aware of something, she is also aware that he is aware that he is aware of it. It is important to point out that this does not imply that Bob is actually aware of it. We note that when considering a relevant view of the form υ   ˆ i we have that υ   ˆ i   ˆ i is relevant and by C1 we have i ∈ I υ   ˆ i hence all players are aware of their own participation in the game.

C4 For every action profile^[2] ( a ) υ   ˆ   υ ˜ = { a j } j ∈ I υ   ˆ   υ ˜ there exists a completion to an action profile ( a ) v = { a j ,   a k } j ∈ I υ   ˆ   υ ˜ , k ∈ I υ ∖ I υ   ˆ   υ ˜ such that

Since a view may consider only some of the players considered by another relevant view, or by the modeler, the payoffs may not be uniquely determined by defining the restricted set of players and actions. The fourth condition requires that the payoffs in a restricted game coincide with payoffs in the larger game with more players by fixing some action profile for these players. We note that for different action profiles of the restricted game we can have different completing actions by players that only appear in the less restricted game^[3]. In other words, a restricted view of the game cannot introduce new payoffs. If one wishes, this condition can easily be generalized to assume missing players are playing mixed strategies, or even correlated strategies, without impact on our results.

These properties are used in all game forms discussed below. While the objects in the game may change (with the addition of histories, types, or game trees) we obtain analogous properties describing the structure of games with restricted reasoning translating conditions C1-C4 to the appropriate setting. Similarly, we consider standard game forms, e.g., finite normal form games, as the building blocks for our games with unawareness. While many of our results extend to a variety of generalizations this simplifies stating our constructions and results in a unified manner across game forms.

2.2 Hierarchies and Types

Before we define the extension to additional game forms, we briefly discuss our choice for describing a game with unawareness as a collection of standard form games – a “hierarchy” of games, rather than a “type space” – a set of type profiles representing each player’s perception of the game and perception of the types of other players. First we note that a representation a la Harsanyi’s type spaces is possible and actually quite readily obtained and is briefly provided below. Moreover, one can also define a universal type space which corresponds to the incomplete information representation we provide in Section 3.2. Hence, the transition from hierarchies to type spaces and back is quite a standard (if tedious) exercise. However, this brings us to a central reason for choosing hierarchies: in the standard incomplete information framework, the set of fundamentals generating the universal type space is fixed. In particular, all players are implicitly assumed to be reasoning with the same set of fundamentals which, in the Harsanyi’s approach, includes the collection of relevant payoff matrices for the fixed and given set of players and their given sets of actions. This makes the universal type space “common knowledge” in the sense that all players would construct the same universal type space if they were modeling the game, see Aumann (1999) for a discussion of this property. In contrast, our starting point is that players may reason using different fundamentals – the building blocks of the game. As such, they would construct different universal type spaces – different from each other and different from the one the modeler uses. Moreover, the players would also attribute different universal type spaces to other players. As such, while we can combine all these to an abstract space as sketched below, working with such a space seems much more difficult than the analysis we did in the examples in the introduction via the hierarchy representation. Working with a type space representation that captures reasoning with differing fundamentals requires extra caution since one needs to figure out at each state how each player perceives the type space. While this is possible, it essentially amounts to reconstructing the hierarchies. Hence, we prefer working directly with hierarchies. This is best exemplified when constructing solution concepts. If we want a player type to play a best response to what other players’ types are playing we need to consider the types that correspond to this player’s type space, which is not the modeler’s type space. In addition, this player will need to evaluate the type spaces that each other player is considering, the spaces that each of them associates to other players (more precisely, the spaces that the first player perceives the other players think others are considering), and so on – recreating the hierarchies once again. Finally, the hierarchies allow us to obtain an “equal perception” principle. In the sense that the modeler and the players are easily shown to model the situation in the same manner (as well as model others’ perceptions and so on). While they may use different fundamentals for reasoning and attribute different fundamentals to others, the rules governing the reasoning, the relationships between these sets of fundamentals and the definition governing the game form all are of the exact same form. Not only are the games the players consider of the same form, the games they perceive other consider are of the same form and all these coincide with the modeler’s game form. This is not a minor issue since, at least in economics, one might consider it desirable that a model used in describing or predicting behavior will not be at odds with its own predictions if the modeled players themselves were to use the exact same modeling approach. When one begins with an abstract modeler type space alone, it is not a priori clear whether this property holds.

We describe a type space representation for normal form games with unawareness with two players that are aware of each other. The generalizations are later described. The type space representation is defined as a product T 1 × T 2 × Λ and mappings τ 1 ,   G 1 ,   τ 2 ,   G 2 such that for i ,   j ∈ { 1 ,   2 } and j ≠ i we have: T i is player i’s type space, Λ is a set of normal form games, τ i : T i → T j and G i : T i → Λ . Hence, each t i ∈ T i , is mapped to a member τ i ( t i ) ∈ T j and a game G i ( t i ) ∈ Λ , these correspond to how type t i views j’s view of the game with unawareness and the game as t i views it. Note that the hierarchy of how i’s type views j’s view of i’s view etc, is captured via iterations of the mappings τ and G.

We now add the conditions that correspond to conditions C1-C4 for the type space representation. Condition C1 will be represented by the requirement that the game G i ( t i ) is a two person game played by i and j. Condition C3 will hold immediately from the type space representation. Condition C4 holds for this specific case where there is no unawareness of players. For condition C2 we will require that the game G j ( τ i ( t i ) ) is obtained from the game G i ( t i ) by (at most) eliminating pure strategies.

We need to show that every game with unawareness corresponds to a hierarchy of some type space and mappings which satisfy the above conditions and that for every type space with mappings as above we have that every pair of types generates a hierarchy that corresponds to a game with unawareness. The first part will be shown when we discuss the canonical representation of games with unawareness later on. The second part follows from noting that for a given type t i , we can define the following hierarchy of games G i ( t i ) ,   G j ( τ i ( t i ) ) ,   G i ( τ j ( τ i ( t i ) ) ) ,   … , which corresponds to the views i ,   i ˆ j ,   i ˆ j ˆ i ,   … respectively. For every pair of types t 1 , t 2 we define G ϕ as the game that contains the union of pure strategies of G 1 ( t 1 ) and G 2 ( t 2 ) (choosing arbitrary payoffs for payoff of action profiles that are not already present in either G 1 or G 2 ). We define the games for all views by extending the above inductively so that G υ ˆ v ˆ υ ‾ = G υ ˆ v ˆ v ˆ υ ‾ and we have a game with unawareness satisfying C1-C4.

This equivalence extends to more than two players without any change in conditions. For the case where players may be unaware of other players we need to modify τ i so that it does not necessarily map to a type profile of all other players, but only the players that the specific type t i is aware of. Similarly, G i ( t i ) will include exactly the players that τ i ( t i ) maps to in addition to player i herself. Finally, we require that for a player j in G i ( t i ) the game G j ( τ i ( t i ) ) is obtained from G i ( t i ) by at most eliminating actions and players and preserving the payoff condition as in condition C4. Thus, the resulting type space and mappings will represent all normal form games with unawareness. Note that while the representation has a fairly simple space structure, the set of fundamentals considered by different types (even of the same player) can be quite different.

For the most part a compact representation of a game with unawareness can be produced. For example, in the first example in the introduction Alice and Bob are both aware that the game is as in (1), but Alice thinks that Bob is unaware of one of her actions and considers the game as in (2). Bob actually realizes that Alice attributes to him this restricted view. The representation of this game with unawareness can be collapsed to three states { x ,   y ,   z } , two states x ,   y corresponding to (1) and the state z to (2). Alice is unaware of the actual state x, at state x she considers state y to be the actual state, at state y Bob would consider state z to be the actual state and he would be unaware of state y. Hence, Alice thinks Bob views the game as (2) while Bob realizes this and actually views the game as (1). For the seminal formal treatment of interactive unawareness and in particular the relationship between models (state space formulation) and the syntax of hierarchies see Fagin and Halpern (1988).

2.3 Repeated Games with Unawareness

Repeated games introduce a dynamic environment where the restricted view of the players may change as the games unfolds. Their view may be widened from directly observing behavior they previously did not consider, from reflecting about behavior, or any other means of discovery. A view can also dynamically narrow from forgetfulness, or by deeming some aspect of the game irrelevant at some point.

As with the normal form games, we consider a collection of repeated games corresponding to relevant views – sequence of viewpoints. This time, however, a player’s limited view may depend on the period in which the game is played and, most importantly, it may also depend on the history of play that this player observed. Hence, the collection of potential viewpoints includes every player at every period conditional on every possible history of actions by a subset of players.

We define repeated games with unawareness as a collection of standard repeated games satisfying the four consistency conditions as above. The first condition accommodates the added dynamic constraint on relevant viewpoints which now refers to specific histories. For example, if Bob cannot reason about a certain action in the first period of the repeated game, he cannot reason in this first period about his, or Alice’s, reasoning in the second period conditional on this action being taken. As such, the viewpoint after the action has occurred is not relevant for views that do not consider that action possible. The third condition is modified to allow for imperfect monitoring and forgetfulness. In these cases a player may not be aware of, or remember, the history of play which requires a modification of the self awareness condition. The other two conditions remain intact.

Let G ( T ) denote the T ≥ 1 repeated game with a stage game G = ( I ,   ∏ i ∈ I A i ,   { u i } i ∈ I ) and payoffs determined by the sum or average of the stage game payoffs (similarly we can consider the infinitely repeated δ-discounted games G ( δ ) ). Let h t ( J ) = ( a 1 ,   … ,   a t ) denote a history of (restricted) action profiles in the first t periods where for every s we have a s ∈ ∏ i ∈ J A i . Hence a s is the action profile for the given subset J ⊂ I . We denote by A i ( h t ( J ) ) the actions taken by i along the history h t ( J ) assuming i ∈ J , i.e. the set { a i s | s = 1 ,   … ,   t } . Let H = { h t ( J ) | t = 0 ,   1 ,   … ,   T − 1   and   ϕ ≠ J ⊂ I } be the set of all histories up to length T − 1 with h 0 ( J ) = ϕ for all J.

Each player after each history of play of a subset of players constitutes a possible viewpoint. The set of possible viewpoints is defined as the collection V = { ( h t ( J ) ,   i ) | h t ∈ H ,   i ∈ I ,   ϕ ≠ J ⊂ I } , i.e., it is each player’s view of the game as a function of the history at the time of observation when considering some of the players in the game. As before, a typical viewpoint is denoted by v ∈ V .

The set of all finite sequences of viewpoints denoted V ‾ = ∪ n = 0 ∞ V ( n ) with V ( n ) = ∏ ​ j = 1 n V and the convention V ( 0 ) = ϕ . As before a finite sequence of viewpoints υ = v 1 , … , v n is associated with a repeated game G υ T = I υ ,   ∏ i ∈ I υ A i υ ,   u i υ i ∈ I υ where I υ is the set of players that v 1 finds that v 2 finds that … that v n is considering, and similarly for the sets of actions A i υ and payoffs u i υ defined for the stage game with action profiles ∏ j ∈ I υ A j υ and repeated T times.

Definition 2. A collection Γ = G υ T υ ∈ V where G υ T are repeated games and V ⊂ V ‾ is a set of relevant views is called a repeated game with unawareness if the following properties hold:

CR1 For every υ ∈ V ,   v = h t J ,   i we have

The viewpoints that are considered relevant from the view υ are exactly those that correspond to players and histories taken from the stage game G υ ( T ) .

CR2 For every υ ˆ   υ ~ ∈ V we have

and

as well as

for every i ∈ I υ ˆ υ ~ .

CR3 If υ   ˆ   v   ˆ     υ ‾ ∈ V where v = ( h t ( J ) ,   i ) then there exists some v ˜ = ( h ˜ t ( J ˜ ) ,   i ) such that

and υ ˆ v ˆ v ˜ … v ˜ ˆ υ ‾ ∈ V .

This refines condition C3 by allowing a viewpoint to consider itself with respect to the histories it can reason about. While a view υ may consider v = ( h t ( J ) ,   i ) because the view finds the history h t ( J ) possible, it will recognize that the decision maker i may not be aware of this history due to imperfect monitoring or forgetfulness. Hence, he might associate himself with other viewpoints such as v ~ = ( h ~ t ( J ~ ) ,   i ) .

CR4 For every action profile a υ ˆ   υ ˜ = a j j ∈ I υ ˆ   υ ˜ there exists a completion to an action profile a υ = a j ,   a k j ∈ I υ ˆ   υ ˜ , k ∈ I υ \ I υ ˆ   υ ˜ such that

The dynamic features of repeated games with unawareness prompt us to consider the properties that relate a player’s past and present views. As demonstrated in the example of a repeated game with unawareness in the introduction, what players learn about the perceptions of others may be quite intricate. This opens up the possibility of modeling a variety of “learning” dynamics, where learning fits a discovery of facets of the game previously not considered, included higher level of perceptions. We find this to be more close to the casual use of the word, to quote Alvin Roth: ”One of the most general things that experiments demonstrate is that subjects adjust their behavior as they gain experience and learn about the game they are playing and the behavior of other subjects.” (Roth 1995 p.327).

Memory For υ = i 1 h t 1 ,   … ,   i k h t k ,   … ,   i n h t n and υ ~ = i 1 h t 1 ,   … , i k h t k + l ,   … ,   i n h t n where h t k + l is a continuation of h t k and such that υ , υ ~ ∈ V ‾ we have that α υ ⊂ α υ ~ . Memory assumes that awareness is monotonic, in the sense that what a player ( i k ) is aware of after a history ( h t k ) he will still be aware of after some continuation of the game. This property states that a player remembers not only actions but also what he was aware of others’ awareness, and it also assumes that others are aware the player remembers. Our results hold both with and without this assumption.

To get a better feel for the potential complexity of repeated games with unawareness consider the game depicted in Figure 1 repeated twice. In period 1 Alice and Bob are viewing each other’s perception at the current period as before: Alice and Bob are aware of the actions { a 1 ,   a 2 ,   a 3 ,   b 1 ,   b 2 ,   b 3 } , and Alice perceives that Bob is unaware of her action a 3 , i.e., she views him as viewing the game as depicted in Figure 2 as in our original example. But how Alice views (at either period) the viewpoint of Bob in period 2 will depend on the realization of play in period 1. If Alice played a 3 in period one, she may safely assume that Bob will remember this action and she will attribute to him the awareness he actually already has. Moreover, if Bob plays b 3 in the first period, Alice might deduce that Bob must have been aware of a 3 in the first period (and if he remembers, in the second period as well) even if she did not choose a 3 in the first period. Hence, Alice’s perception of Bob’s awareness may change not only from the revelation of actions that were considered secretive, but also from behavior that may best be explained by a different scope of awareness Alice should attribute to Bob. In particular, Alice here may realize that her perception was limited (she did not consider Bob considering a 3 ) and revise her perception. In this case, Alice may actually reason at period 1 about how she would reason in period 2 about Bob’s reasoning in period 1 if she observes b 3 . The important feature of the game form illustrated here is that the choice of relevant views may depend on how perception changes with observed behavior – a choice we usually associate with a solution manifests here in the game form. For more on this game and how communication impacts strategic interaction with unawareness see Feinberg (2007).

2.4 Incomplete Information Games with Unawareness

Incomplete information games with unawareness allow us to model uncertainty about the awareness of players in addition to uncertainties about the payoffs as well as high order uncertainties about both. They also allows us to model views that do not consider all possible types. As with other game forms we begin with the modeler’s view which begins with a standard Bayesian game with a prior G ( B ) = ( I ,   ∏ i ∈ I A i ,   Θ 0 × ∏ i ∈ I Θ i ,   P ,   { u i } i ∈ I ) where I is a finite set of players, A i is player i’s finite actions set, Θ = Θ 0 × ∏ i ∈ I Θ i is a set of type profiles where Θ i is the set of player i’s types and Θ 0 is the set of states of nature, the u i ’s are the players’ utilities defined for a realization of Θ and action profiles, and P is a probability distribution over Θ . The distribution P is the probability over the type profiles as seen by the modeler. Recall that in a standard incomplete information game each type has a distribution over the other players’ type profiles. This will be captured in our setting when we describe the game as viewed by each type which will include a probability distribution corresponding to that type’s beliefs.

The set of viewpoints is the set of all possible types V = ∪ i ∈ I Θ i and a typical viewpoint is denoted by v = θ i ∈ V . The set of all finite sequences of viewpoints is V ‾ = ∪ n = 0 ∞ V ( n ) . A finite sequence of viewpoints υ = v 1 ,   … ,   v n is associated with a game G υ B = I υ ,   ∏ i ∈ I υ A i υ ,   Θ 0 υ × ∏ i ∈ I υ Θ i υ ,   P υ ,   u i υ i ∈ I υ where I υ is the set of players that v 1 finds that v 2 finds that … that v n is considering, and similarly for the sets of actions A i υ , states of nature Θ 0 υ , types Θ i υ , with Θ υ = Θ 0 υ × ∏ i ∈ I υ Θ i υ , the distribution P υ over the viewed type space Θ υ = Θ 0 υ × ∏ i ∈ I υ Θ i υ , and payoffs u i υ defined for the viewed states of nature, types and action profiles Θ 0 υ × ∏ i ∈ I υ Θ i υ × ∏ j ∈ I υ A j υ . The conditions defining the extension to unawareness are almost identical to those used for the normal form games.

Definition 3. A collection Γ = G υ B υ ∈ V where G υ B are games as above and V ⊂ V ‾ is a set of relevant views is called an incomplete information game with unawareness if the following properties hold:

CI1 For every υ ∈ V , v = θ i ∈ Θ i we have

CI2 For every υ   ˆ υ ~ ∈ V we have

and

as well as

for every i ∈ I υ   ˆ   υ ~ .

CI3 If υ   ˆ   v   ˆ   υ ‾ ∈ V we have

and υ   ˆ   v   ˆ   v   ˆ   υ ‾ ∈ V .

CI4 For every state (nature and type profile) and action profile pair θ ,   a υ   ˆ   υ ~ ∈ Θ 0 υ   ˆ   υ ~ × ∏ i ∈ I υ   ˆ   υ ~ Θ i υ   ˆ   υ ~ × ∏ j ∈ I υ   ˆ   υ ~ A j υ   ˆ   υ ~ there exists a completion to a pair θ ,   a υ ∈ Θ υ × ∏ j ∈ I υ A j υ that agrees with the appropriate coordinates of θ , a υ   ˆ   υ ~ such that

Note that the notion of a type in an incomplete information game with unawareness differs from a type in a standard incomplete information game. The difference is that the beliefs a type has over the type space (as he perceives it) need not be held in common knowledge. Moreover, types are allowed to perceive different type spaces. For example, a type θ i may have a belief P θ i yet type θ j may conceive type θ i ’s belief to be different, P θ j θ i ≠ P θ i and even defined on a different space.

It is natural to ask whether an analog to Harsanyi’s consistency condition – the common prior assumption – can be found for these games with unawareness. The obvious prior candidate is the modeler distribution P = P ∅ , however, since a type θ i may be unaware of the whole space Θ his distribution P θ i may be defined on a different set of types ( Θ ) ( θ i ) , and if θ i is unaware of some players then ( Θ ) ( θ i ) may not be a subset of Θ but rather a subset of a projection. Although this implies that P θ i cannot be merely a conditional of P, it indicates that a projection of the prior might do. Letting P θ i be exactly the conditional probability over Θ ( θ i ) of the marginal of P with respect to Θ 0 × ∏ j ∈ I θ i Θ j , and extending this definition to iterated relevant views provides a candidate for an extended common prior condition.

Definition 4. We say that consistency (in the sense of Harsanyi) holds for an incomplete information game with unawareness if for every relevant υ   ˆ υ ~ ∈ V we have

Note that this is the marginal of P υ with respect to Θ 0 υ × ∏ j ∈ I υ   ˆ   υ ~ Θ j υ and then conditional on the types from the perspective υ   ˆ   υ ~ , i.e., conditional on the potential subsets of types considered Θ 0 υ   ˆ   υ ~ × ∏ j ∈ I υ   ˆ   υ ~ Θ j υ   ˆ   υ ~ . Copic and Galeotti (2005) have independently modeled incomplete information games with unawareness in quite a similar manner. The difference is that they consider players and actions as commonly known and modeled unawareness of types and their beliefs. While our model above is more general, we find it reassuring that the structure of the two definitions is essentially the same.

2.5 Extensive Form Games with Unawareness

The final, and in some sense most general, class of games we consider are extensive form games. This game form captures both uncertainties and dynamics. An extensive form game is composed of a game tree capturing decision points, actions, nature moves, information sets, probabilities for nature moves and payoffs. An extensive form game is denoted G ( D ) = ( ( W ) ,   I ,   A 0 × ∏ i ∈ I A i ,   { F i } i ∈ I ,   P ,   { u i } i ∈ I ) where W , ≺ is a finite tree (infinite games vary in definition and unawareness can be extended accordingly) with a disjoint union of vertices (a partition) W = V 0 ∪   ∪ i ∈ I V i ∪ Z where V 0 is the set of natures moves, V i denotes the set of player i’s decision points and Z is the set of terminal vertices and the order w ' ≺ w denotes that w ' occurs before w on the tree.

Denote by P r e d ( w ) and S u c c ( w ) the (immediate) predecessor, and respectively successor, of w – formally it is the maximal vertex smaller than w and the minimal larger than w respectively. A pair e = ( w , S u c c ( w ) ) of a vertex and its successor is called an edge and the set of edges emanating from w is denoted E ( w ) . We assume that every two vertices in the tree are connected with a finite path – a finite sequence of edges. Hence, every vertex has a (single) predecessor except for the root that has none and the set of terminal vertices Z ⊂ W is the set of vertices that have no successor. The set of players is I. An edge e = ( w , w ' ) belongs to player i, resp. Nature, when w ∈ V i , resp. w ∈ V 0 . The mappings A i ( w , w ' ) are defined for player i’s edges, w ∈ V i , w ’ = S u c c ( w ) , A 0 ( w , w ' ) when ( w , w ' ) is a nature move, and associate an action with each edge in E ( w ) . The partitions F i of the sets V i correspond to the information sets of player i and f i ( w ) ∈ F i denotes the partition member containing a vertex w ∈ V i . We require that for every w the function A i ( w , · ) of the successors of w is one to one, that for every pair w ≠ w ’ the set of values that A i ( w , · ) and A i ( w ' , · ) obtain are disjoint unless w ' ∈ f i ( w ) ∈ F i in which case they are identical. The mapping P associates a probability distribution over the edges following each of nature’s vertices. For every w ∈ V 0 we denote by P ( w ) the probability distributions over E ( w ) . Finally, u i : Z → R are the utilities of players defined for terminal vertices.

The set of viewpoints is the players’ set of decision points V = ∪ i ∈ I V i and a typical viewpoint is denoted by v. The set of all finite sequences of viewpoints is  V ‾ = ∪ n = 0 ∞ V ( n ) . A finite sequence of viewpoints υ = v 1 , … , v n is associated with an extensive form game G υ D = W υ , ≺ ,   I υ ,   A 0 υ × ∏ i ∈ I A i υ ,   F i υ i ∈ I υ ,   P υ ,   u i υ i ∈ I υ where W v = ( V 0 ) v   ∪   ∪ ​ i ∈ I υ ( V i ) v   ∪   Z v and similarly for all other ingredients of the game in accordance with the description above. We use the same ordering ≺ since W υ will be subsets of W.

Definition 5. A collection Γ = G υ D υ ∈ V where G υ D are games as above and V ⊂ V ‾ is a set of relevant views is called an Extensive Form Game with unawareness if the following properties hold:

CD1 For every υ ∈ V , v ∈ V i we have

CD2 For every υ   ˆ   υ ~ ∈ V we have

as well as for all i ∈ I υ   ˆ   υ ~ , w ∈ V i υ   ˆ   υ ~

and

for the unique successor w '' of w in W v such that w '' ≺ w ' , where w ' is the successor of w in W υ   ˆ   υ ~ .

CD3 If υ   ˆ   v   ˆ   υ ‾ ∈ V with v ∈ V i then we have f i v ∩ V i υ   ˆ v ≠ ϕ and for every v ~ ∈ f i v ∩ V i υ   ˆ   v we have

and υ ˆ v ˆ v ~ … v ~ ˆ υ ‾ ∈ V .

The third condition states that a viewpoint must consider its information set to be relevant. Moreover, the viewpoints it considers in its information set are assumed to find themselves relevant and model the game in the same manner. Otherwise, a player at an information set would be able to distinguish decision points based on differing views at the decision points.

CD4 Let υ ˆ υ ‾ ∈ V . For every terminal vertex w ∈ Z υ   ˆ   υ ‾ there exists a vertex w ' ∈ Z υ such that w ≺ w ' and

Note that throughout our constructions and conditions the ordering ≺ for a subset of the histories is inherited from the ordering in the original standard game. As with incomplete information games we did not constrain the subjective probabilities that a viewpoint, or a view, associates with nature moves. If a restricted view of the game omits some nature moves one may still impose an analog for common priors:

Definition 6. We say that Harsanyi consistency holds for a extensive form game with unawareness representing an incomplete information game with unawareness, if for every relevant υ ∈ V we have at every w ∈ V 0 υ that

We note that the definition of condition CD3 is stronger than the repeated games version CR3 as it requires that a viewpoint not only see itself as a viewpoint in the same information set and agree with it, but also that it will agree with all its viewpoints in the restricted information set. This definition follows the interpretation that every player at an information set perceives it as representing indistinguishable information and indistinguishable awareness. In other words, in an information set of a game with unawareness the relevant player will perceive all their decision points to have the same perception of the game.

Our main structural result states that games with unawareness are consistent in the sense that from every view of the game is seen as a game with unawareness satisfying the exact same conditions as the modeler’s game.

Consider a game with unawareness Γ = G υ ⋅ υ ∈ V where G v ( ⋅ ) has one of the four forms: normal, repeated, incomplete information or dynamic. For every relevant υ ∈ V we define the relevant views as seen from v as: V υ = υ ~ ∈ V ‾ | υ ˆ υ ~ ∈ V . For each relevant view υ ~ ∈ V υ we define the game G υ ~ υ ⋅ = G υ   ˆ   υ ~ ⋅ and the game with unawareness as seen from v is defined as Γ υ = G υ ~ υ ⋅ υ ~ ∈ V υ .

The proof of this as well as all other propositions in this paper appears in the Appendix.

We note that this game form allows for imperfect recall, i.e., players may not only discover aspects of the game they were not aware of, they can also forget past actions. If one would rather maintain perfect recall two extra conditions are required. First the actual game with all the players and actions must be a game of perfect recall. Furthermore, at every decision point a player must consider a game that includes the game they considered in the past, i.e., no player may have their view of the game contract over time. Imposing the latter condition on higher order views (everyone views everyone’s view … to include all past views) will allow the conditioning on perfect recall at any reasoning level. Note that this condition still allows a player to revise their view of what other players view of the game is. For example, Alice can observe an action by Bob that may lead her to think that Bob is less aware than she previously thought. This does not imply that she believes Bob forgot, it is a revision of Alice’s view of Bob’s view at a given decision point for Bob. Once she revises this view, she may well assign a restricted view for Bob in future decision points. Still, she will assume that Bob has perfect recall as she revises all of his views to be more restrictive.

3.1 Rationalizability and Nash Equilibrium in Normal Form Games with Unawareness

In order to define the solutions for games with unawareness we need to associate behavior with each possible view.

Definition 7. Let Γ = G υ υ ∈ V be a normal form game with unawareness. An extended strategy profile ESP in this game is a collection of strategy profiles σ υ υ ∈ V where σ υ is a strategy profile in the game G υ such that for every υ ˆ v ˆ υ ‾ ∈ V we have

in the sense that the same pure strategies are assigned the same probabilities in the two games G υ and G υ   ˆ   v , as well as

The first condition requires that the strategy that the view v associates with player v in the game G υ is the same strategy that the view v finds the player playing in the game as he is seen to see it, i.e., in the game G υ   ˆ   v . We abuse the notation in the equalities in the definition since σ v υ and σ v υ   ˆ   v may reside in different pure strategy product spaces, however the whole point is that one is a product of subsets of the other where the support of the strategies must lie and the probabilities coincide. Consider, for example our repeated game from the introduction, Alice perceives that Bob is not aware of non-cooperation hence she will not consider he would play it as she considers his strategy in the game with non-cooperation actions that she is aware of. The interpretation is that whenever a strategy is assigned to a player from a given perspective while taking into account the player’s perception, that player is indeed assumed to be playing the strategy from that perspective. The second condition follows the same logic behind condition C3 in stating that the behavior associate by a player to the games he views is identical to the behavior he reasons about himself associating to games, and that this principle holds from every view. In other words, Bob’s strategy associated with the game that Alice perceives that Bob perceives is the strategy that Bob is assumed to play in the game as Alice perceives it. In addition Bob’s view of his own view of the strategy coincides with his view of the strategy and this is commonly understood at every view.

It is worthwhile noting that the definition of an extended strategy restricts the behavior of players to actions they are aware of. In the definition of a game with unawareness we allowed the possibility that a player may have an action he is unaware of, i.e., a view υ may perceive a player v as having an available action a, i.e., a ∈ A v υ , while at the same time v may perceive that v is unaware of a, i.e., a ∉ A v υ   ˆ   v . In this case the right-hand side of (38)   is defined on a strictly smaller set of actions and we assume that the right-hand side support is in that set, hence the ESP is defined such that σ v υ assigns 0 probability to the pure strategy a when the player is unaware of it.

We begin by defining rationalizability in games with unawareness. As expected, rationalizability corresponds to playing a best response in the perceived game to perceived strategies that are themselves best responses in how it is perceived the corresponding players view the game, and so on. This extends rationalizability from normal form games to normal form games with unawareness.

Definition 8. An ESP σ υ υ ∈ V in a game with unawareness is called extended rationalizable if for every υ ˆ v ∈ V we have that σ υ υ is a best response to σ − υ υ   ˆ υ in the game G υ   ˆ υ .

The principle governing the extension of NE to games with unawareness follows the epistemic foundation of the solution concept. A NE requires rationalizability – players play a best response to conjectures, and some form of truth – knowledge of conjectures, or agreement on strategies. These correspond in our setting to strategies that are best responses at every view, and to strategies that coincide when the views of the game coincide, respectively. The first property corresponds to rationality in the sense of playing a best response to conjectures. The second property requires that the conjectures, or best responses, are the same when reasoning about the same game. In other words, when players have the same perceptions about the game (with unawareness) they share the same conjectures on behavior – agreement on strategies.

For a game with unawareness two views υ , υ ‾ share the same perception of the game if they agree on how all other views consider the game, i.e., they consider the same game with unawareness Γ υ = Γ υ ‾ .

Definition 9. An ESP σ υ υ ∈ V in a game with unawareness is called an extended Nash equilibrium (ENE) if it is rationalizable and for all υ   ˆ ,   υ ‾ ∈ V such that Γ υ = Γ υ ‾ we have that σ υ = σ υ ‾ .

Note that an ENE assigns the same behavior in games corresponding to the concatenation of views once the perceptions of the game coincide, i.e., Γ υ = Γ υ ‾ implies that for all υ ~ such that υ ˆ υ ~ ∈ V we have ( σ ) υ   ˆ   υ ~ = ( σ ) υ ‾   ˆ   υ ~ . This follows from noting that Γ υ = Γ υ ‾ implies that Γ υ υ ~ = Γ υ ‾ υ ~ .

We justify the use of the term “extended” with the following result. This result states that when all views see the game in the exact same manner – there is no unawareness – then the extended solutions coincide with their standard counterparts for the normal form game at hand. More generally, at every view such that the game is seen to have no unawareness the extended solution coincides with the standard one.

While a game with unawareness may correspond to an infinite collection of games, the structure does support the existence of an equilibrium:

Condition C2 guarantees that every view of how a viewpoint perceives the game is a restriction of the original view of the game. However, a game with unawareness Γ may still incorporate an infinite number of differing views of the game with unawareness, i.e., the set of games with unawareness Γ υ υ ∈ V could have an infinite number of distinct members:

Example 1. Consider three players denoted 1,2,3 and let G be a normal form game where player 1 has three actions and players 2 and 3 have a single action each. Let F be a normal form game obtained from G by removing one of player 1’s actions, and let E be a normal form game obtained from F by removing one of the two remaining actions of player 1. We define the game with unawareness Γ as follows.