Reasoning about Social Choice and Games in Monadic Fixed-Point Logic

07/22/2019 ∙ by Ramit Das, et al. ∙ Indian Institute of Technology Kanpur 0

Whether it be in normal form games, or in fair allocations, or in voter preferences in voting systems, a certain pattern of reasoning is common. From a particular profile, an agent or a group of agents may have an incentive to shift to a new one. This induces a natural graph structure that we call the improvement graph on the strategy space of these systems. We suggest that the monadic fixed-point logic with counting, an extension of monadic first-order logic on graphs with fixed-point and counting quantifiers, is a natural specification language on improvement graphs, and thus for a class of properties that can be interpreted across these domains. The logic has an efficient model checking algorithm (in the size of the improvement graph).

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1 Introduction

A logical study of game theory aims at exposing the assumptions and reasoning that underlies the basic concepts of game theory. This involves the study of individual, rational, strategic decision making between presented alternatives (in the non-cooperative setting). One potential form of reasoning in such a situation is to envisage all possible strategic choices by others, consider one’s own response to each, then others’ response to it in their turn, and so on

ad infinitum, with Nash equilibrium representing fixed-points of such iteration..

Such reasoning, which we might call improvement dynamics, is similar to but distinct from rational decision making under uncertainty; it is also similar to but distinct from epistemic reasoning. The former is about optimization, selecting the ‘best’ option in light of one’s information; the latter is about ‘higher order information’ involving information about others’ information etc. Improvement dynamics intends to yield the same end results as these, but operates at a more operational, computational level, and reasoning about it can be seen as reasoning at the level of computations searching for equilibria. In this sense, logic is seen as a succinct language for describing computational structure, rather than as a deductive system of reasoning by agents. In spirit, the role of such logics is similar to that of logics in descriptive complexity theory. If we were to talk of the descriptive complexity of game theoretic equilibrium notions, it would need to account for the implicit improvement dynamics embedded in the solution concept.

Interestingly, several contexts in social choice theory embed such improvement dynamics as well. When we aggregate individual choices or preferences into social choices / preferences, or decide on social action (like resource allocation) based in individual preferences, once again we see implicit improvement dynamics. If a particular profile of voter preferences yields a specific electoral outcome, one can consider a voter announcing a revised (and altered) preference to force a different outcome. Two agents might exchange their allocated goods to move to a new allocation, if they perceive advantage in doing so. Again, these can be seen as offers and counter-offers, perhaps leading to an equilibrium, or not. Some of these situations involve individual improvements, some (like pairs of agents swapping goods) involve coalitions, but they have the same underlying computational structure.

In this paper, we suggest that monadic fixed-point logic (with counting) is a suitable language for reasoning about this computational structure underlying games and social choice contexts. This is an extension of first order logic with monadic least fixed-point operators and counting. In this, we follow the spirit of descriptive complexity, where extensions of first order logics describe complexity classes. Formulas offer concise descriptions of reasoning embedded in improvement dynamics.

Why bother? When we have a common language across contexts, we can employ a form of reasoning common in one (say normal form games) in another (say fair resource allocations) and thus transfer results and techniques. We show that the idea of improvement under swaps corresponds to certain form of strong equilibria and coalitional improvement in games. Dynamics in iterated voting again correponds to improvement dynamics in games. In such cases when the structures studied possess interesting properties such as the finite improvement property or weak acyclicity we get certificates of existence of equilibria. Interesting subclasses of games (such as potential games) possess such properties and by “transfer” we can look for similar subclasses in social choice contexts, and vice versa.

The choice of monadic fixed-point logic is also motivated by the fact that it admits an efficient model checking algorithm. Monadic least fixed point operator, iterating over subsets of strategy profiles, suffices for improvement dynamics. Counting can help us constrain paths succinctly: though counting is first order expressible, such expression would be prohibitively long.

Thus the contribution of this paper is modest and simple. The reasoning discussed is familiar, that of improvement dynamics in normal form games, and expressing this in monadic fixed-point logic with counting. In the process, we can study the same properties in different contexts, such as normal form games, fair allocations and electoral systems. We also present a model checking algorithm for the logic.

Logic and game theory. Various logical formalisms have been used in the literature to reason about games and strategies. Action indexed modal logics have often been used to analyse finite extensive form games where the game representation is interpreted as models of the logical language [9, 6, 7]. A dynamic logic framework can then be used to describe games and strategies in a compositional manner [34, 20, 35] and encode existence of equilibrium strategies [23]. Alternating temporal logic (ATL) [2] and its variants [24, 42, 13] constitute a popular framework to reason about strategic ability in games, especially infinite game structure defined by unfoldings of finite graphs. These formalism are useful to analyse strategic ability in terms of existence of strategies satisfying certain properties (for example, winning strategies and equilibrium strategies). Some of the above logical formalism are also able to make assertions about partial specifications that strategies have to conform to in order to constitute a stable outcome. In this work we suggest a framework to reason about the dynamics involved in iteratively updating strategies and to analyse the resulting convergence properties. [8] consider dynamics in reasoning about games in the same spirit as ours and describe it in fixed-point logic. But crucially, the dynamics is on iterated announcements of players’ rationality, and belief revision in response to it. Moreover, they discuss extensive form games rather than normal form games. However, they do advocate the use of the fixed-point extension of first order logic for reasoning about games.

Monadic least fixed point logic (MLFP) is an extension of first-order logic which is well studied in finite model theory [39]. It is a restriction of first order logic with least fixed point in which only unary relation variables are allowed. MLFP is an expressive logic for which, on finite relational structures, model checking can be solved efficiently [16]. It is also known that MLFP is expressive enough to describe various interesting properties of games on finite graphs. MLFP can also naturally describe transitive closure of a binary relation which makes it an ideal logical framework to analyse the dynamics involved in updating strategies and its convergence properties. When is a formula with one first order free variable, asserts that the number of elements in the domain satisfying is at most . Clearly, this is expressible in first order logic with equality, but at the expense of succinctness. In the literature on first order logic with arithmetical predicates [38], it is customary to consider two sorted structures to distinguish between domain elements and the counts, but since our domain elements are always profiles, there is no need for such caution.

It well known that a variety of contexts in the mathematical social sciences can be formulated in terms of improvement dynamics leading to equilibria (of some kind). Our observation here is that the deployment of the MLFPC logic can help to unify algorithmic techniques across these contexts. Rather than devise an algorithm for each problem of this kind, definability in MLFPC can at once give a uniform algorithm, which could then be fine-tuned. Admittedly when we present contexts as diverse as normal form games, allocations in social choice theory or voting rules, all in one uniform framework, we only get a broad-strokes description of the models, and the literature on these contexts vary widely in details. We hope to convince the reader that a priori, the MLFPC has sufficient expressiveness to capture interesting variations. Our hope is to delineate the logical resources needed to express the variations, but that will require more work ahead.

2 The improvement graph structure

Improvement dynamics is a natural notion to study in the context of any situation involving strategic interaction of agents. In this section we formalise this dynamics in terms of the data structure called improvement graphs. We consider three specific application domains: strategic form games, voting theory and allocation of indivisible items. We show how improvement graphs can be interpreted in these applications and argue that the analysis of the structure acts as the basis for reasoning about strategic interaction.

Let denote the set of agents. Each agent is associated with a finite set of choices . A profile of choices (one for each agent) induces an outcome in the strategic interaction. Let denote the set of all choice profiles, denote the set of all outcomes and denote the outcome associated with the profile . Each agent is associated with a preference ordering over the outcome set: . This ordering induces a preference ordering over profiles as follows: for and , if . For a choice profile , we use the standard notation to denote the tuple arising from in which the choice of agent is removed.

The associated improvement graph is the directed graph where and . We will denote the triple by . The edge relation satisfies the condition: for and , we have if , and . An improvement path in is a maximal sequence of profiles such that for every there is a player such that . Note that here we use deviation by a single player to define the improvement graph. We could easily extend the definition to deviation by a subset of players, this interpretation might be more relevant in certain domains.

2.1 Strategic form games

A strategic form game is given by the tuple where the set of strategies for agent can be viewed as its set of choices. For , the function associates an outcome to every strategy profile. In this paper, we consider only finite strategic form games. The notion of best response and Nash equilibrium are standard: is best response to if for all , ; is a Nash equilibrium if for all , is best response to . Existence of Nash equilibrium and computation of an equilibrium profile (when it exists) are important questions in the context of strategic form games.

Given a strategic form game , let denote the improvement graph associated with (as defined above). Improvement paths in correspond to maximal sequence of strategy profiles that arise by allowing players to make unilateral profitable deviations that result in improving their choice according to their preference ordering. We say that a game has the finite improvement property (FIP) if every improvement path in is finite [32]. In an improvement path, if each edge in the sequence is the best response of agent to then is called a best response improvement path. We can analogously define the finite best response property (FBRP) if every best response improvement path is finite. FIP not only guarantees the existence of Nash equilibrium, but also ensures the stronger property that a decentralised local search mechanism convergences to a equilibrium outcome. Various natural classes of resource allocation games like congestion games [37], fair cost sharing games and restrictions of polymatrix games [3] are known to have the FIP.

A weakening of FIP was proposed by Young [43] which insists on the existence of a finite improvement path starting from any initial strategy profile. Classes of strategic form games that satisfy this property are called weakly acyclic games. Note that weak acyclicity ensures that a randomised local search procedure almost surely convergence to an equilibrium outcome [29]. Examples of classes of games which has this property include congestion games with player specific payoff functions [31], certain internet routing games [17] and network creation games [26].

As we can see, the improvement graph presents a data structure for analysing normal form games. It captures the epistemic reasoning underlying player choices: if I were to consider a particular profile of choices by all of us, I would rather choose another strategy to improve by my payoff; in that case, agent would revise her choice; and so on, unless we reach a profile from where none of us has any reason to deviate. Such reasoning is closely related to pre-play negotiations studied by game theorists.

2.2 Voting systems

Consider an electorate consisting of a set of voters and a set of candidates. Let be a voting rule that considers the preference of each voter over the candidates and chooses a subset of winning candidates of size (since among candidates have to be elected). The strategy sets for all voters are the same . The outcome set is . The voting rule specifies which candidates win given the complete preferences of all voters. We assume that each voter has a preference ordering over the outcome set . Thus the voting system can be given by the tuple .

The improvement graph associated with is as before: where , the set of strategy profiles of voters; is the improvement relation for voter , given by: if , and for all , .

Voting equilibria have been studied by Myerson and Weber [33]. In general, one speaks of the bandwagon effect in an election if voters become more inclined to vote for a given candidate as her standing in pre-election polls improve, or the underdog effect, if voters become less inclined to vote for a candidate as her standing improves. Myerson and Weber suggest that equilibrium arises when the voters, acting in accordance with both their preferences for the candidates and their perceptions of the relative chances of candidates in contention for victory, generate an election result that justifies their perceptions. Note that the improvement path again gives us the possibility of ‘interaction’ arising from voter preferences, and we can analyse this in the context of specific voting rules.

Given that agents may have incentive to strategically misreport their preferences, it is natural to study the convergence dynamics when voting is modelled as a game. Iterative voting [27, 36, 30] is a formalism that is useful to analyse the strategic dynamics when at each turn a voter is allowed to alter her vote based on the current outcome until it converges to an outcome from which no voter wants to deviate. In general, the outcome of iterative voting may depend on the order of voters’ changes. Again, voters act myopically, without knowing the others’ preferences. This dynamics is again reflected by the improvement path as discussed here and sink nodes correspond to Nash equilibria. Thus given a voting rule, it is natural to ask what equilibria are reachable from a given vote profile.

2.3 Allocation of indivisible goods

An important problem often studied in economics and computer science is the allocation of resources among rational agents. This problem is fundamental and has practical implications in various applications including college admissions, organ exchange and spectrum assignment. In this paper, we consider the setting where there are agents and a set of indivisible items. An allocation such that and for all , , . In the most general setting, each agent has a preference ordering over the allocations. Thus an instance of an allocation problem can be specified as a tuple . Let denote the set of all allocations. In this setting, each allocation can be viewed as defining an outcome and agents have a preference ordering over such outcomes. In a typical allocation problem, it is often assumed that the preference ordering for each agent depends only on the bundle of items assigned to agent . A special case of the above setting is when (i.e. the number of agents and the items are the same) and is required to be a bijection. An instance of such an allocation problem along with an initial allocation defines the well known Shapley-Scarf housing market [40]. If agents are allowed to exchange items with each other, stability of allocation is a very natural notion to consider. Core stable outcomes are defined as allocations in which no group of agents have an incentive to exchange their items as part of an internal redistribution within the coalition. The improvement graph structure can capture the dynamics involved in such a sequence of item exchange in a natural manner. The associated improvement graph can be defined as where . Since the deviation involves exchange of goods among a subset of players (rather than a unilateral deviation by a single player), the edge relation is indexed with a subset . That is, for and , we have if for all , , and for all , .

A finite path in the improvement graph corresponds to a finite sequence of exchanges that converge to a stable outcome. An important question is whether stable allocations always exist and whether a finite sequence of exchanges can converge to such an allocation. For the housing market, it is known that a simple and efficient procedure often termed as Gale’s Top Trading Cycle, can compute an allocation that is core stable. The allocation constructed in this manner also satisfies desirable properties like strategy-proofness and Pareto optimality.

The question of whether decentralised swap dynamics converges to stable allocation has been studied in various related models of resource allocation. [15] analyses optimality in the setting where pairs of agents exchange items or services. The influence of neighbourhood structures in terms of item exchange and its influence on convergence to stable and optimal allocations is studied in [21, 14]. Exchange dynamics with restricted preferences orderings are considered in [41, 18].

Note that the improvement graph here is different from the ones we discussed earlier in a crucial sense. When agents in swap goods, the allocation for other players outside is unaffected. If each agent’s preference ordering depends only on the valuation of the bundle of items that the agent is allocated then their satisfaction is unchanged. However, agent may swap goods with and then use some the goods acquired to make a swap deal with thus leading to interesting causal chains. In effect the entire space of allocations may be tentatively explored by the agents and the interesting question is whether they settle to an ‘equilibrium’. Again the improvement path offers interesting dynamics, and we can ask whether from any allocation, we can reach one where no agent wishes to make a swap. In the case where the preference orderings depend on externalities [11, 19] the situation is similar to what we observed in the context of strategic form games. Apart from stability, notions of fairness like envy-freeness, propotionality and maximin share guarantee are also well studied in the context of allocation of indivisible items [12, 10]. Analysis of the improvement graph is also useful in the context of fairness notions. Existence of a finite improvement path terminating in a fair allocation would indicate the possibility of convergence to a fair allocation in terms of an exchange dynamics.

2.4 Remark

We have set up an improvement graph from three different models used in the mathematical social sciences. This is of course so that a uniform logical language can be used to specify properties of all these. One natural alternative to consider would be to translate all the models into one, games being the natural choice, and then induce the improvement graph over the defined model. This is certainly possible, but in general this can lead to an increase in the size of the graph. Moreover since we hope to use MLFPC not only to unify these contexts but also differentiate them (in terms of logical resources needed), such a reduction would not be helpful.

3 Monadic fixed-point logic with counting

In this section we present the monadic fixed-point and counting extension of first order logic interpreted over improvement graphs. The use of the fixed-point extension is motivated by the fact that we wish to express properties like acyclicity of the graph, which is not first-order expressible. As we will see below, we need the fixed-point quantifier to range only over collections of nodes, and hence monadic fixed-point quantifiers suffice for our purpose. The counting extension helps us to count nodes in a subgraph, or along a path; this helps us express notions like fairness of schedules, which is of relevance in specifying improvement dynamics.

An alternative formalism would be the transitive closure extension of first order logic. But as Grohe has shown [22], monadic lfp logic is strictly less expressive than transitive closure logic, and hence we prefer a minimal extension of first order logic that serves our purposes. Note that the counting extension does not add expressivess but only succinctness. This is of use when we discuss concurrent deviation by a subset of players.

3.1 MLFPC Syntax

Let be a first order relational vocabulary. Let be a sequence of monadic relation symbols, such that for each , . These are the second order fixed-point variables of the logic.

The set of all MLFPC formulas, , can be defined inductively as follows:

  • Every FO-atomic formula over is an MLFPC formula.

    the set of first order free variables in . the set of all relation symbols occurring in ; .

  • If are MLFPC formulas then so are, , and .

    and .

  • If is a MLFPC formula, and , then so are , and .

    .

  • If is an MLFPC formula, , , and and occurs positively in , then is an MLFPC formula.

    .

The restriction to positive second order variables in the lfp operator is essential to provide an effective semantics to the logic. It is a standard way of ensuring monotonicity, given that we do not have an effective procedure to test whether a given first-order formula is monotone on the class of finite -structures [28].

It should be noted that the use of positive second order variables in no way restricts us to contexts where equilibria are guaranteed to exist. Equilibria are given by graph properties, and these variables allow us to collect sets of vertices monotonically.

3.2 MLFPC Semantics

To interpret formulas, we extend -structures with interpretations for the free first order and second order variables (the latter from the given sequence ). Let be a -structure, which has domain A. The semantics for the first order formulas of MLFPC are standard. The semantics for the count and the fixed point operator are given below.

where let, , , is an operator such that, , ,

The lfp quantifier induces an operator on the powerset of elements on the structure ordered by inclusion. The positivity restriction ensures that the operator is monotone and hence least fixed-points exist.

3.3 Properties

Since the models of interest are improvement graphs, first order variables range over nodes in the graph, monadic second order variables range over subsets of nodes and the vocabulary consists of binary relations , where . When and , we will simply write the relation as . We write formulas of the form to denote .

We now write special formulas that will be of interest in the sequel.

Now consider the formulas interpreted over improvement graphs of normal form games. refers to the set of sink nodes, and these are exactly the Nash equilibria of the associated game. The sentence is true exactly when the improvement graph is acyclic and hence such games have the finite improvement property (as every improvement path is finite). To see that the sentence captures acyclicity, note the action of the lfp operator: at the zeroth iteration, we get all nodes with in-degree ; we then get all nodes which have incoming edges from nodes whose in-degree is ; and so on. Eventually it collects all nodes through which no path leads to a cycle. Since the sentence applies to every node, we infer that the graph does not contain any cycle.

For weak acyclicity, we require that there exists a finite improvement path starting from every node. Again the lfp operator picks up sink nodes at the zeroth iteration, then all nodes that have a sink node as successor, and so on. Eventually it collects all nodes that start finite improvement paths. The sentence asserts that every node has this property.

Note that these formulas capture equilibria, FIP and weak acyclicity on improvement graphs per se, irrespective of whether they arise from normal form games, voting systems, or resource allocations. But there were some differences in the way these graphs were generated, and we turn our attention to these differences.

In improvement graphs of resource allocation systems, we defined the edge relation to be , where , if for , and for all , . Note that an analogous definition in the case of normal form games leads us naturally to a concurrent setting, and we get what are called -equilibria in coordination games.

Let and be strategy profiles in a normal form game, . Define when for , and for all , . That is, with the choices of the other agents fixed, the coalition of agents in can coordinate their choices and deviate to get a better outcome. Thus -equilibria are nodes from which no coalition of at most agents can profitably deviate. We can then define a coalitional improvement path, where at each step a coalition of at most agents deviate, which leads us further to a -FIP. [4] shows that a class of uniform coordination games has this property.

Note that the disjunctions are large, exponential in . Since we have a counting operator, we could add further structure to nodes, prising out the individual strategies of players and then use the counting quantifier over these to get a succinct formula, linear in .

Observe that these formulas apply to coalitions of -agents performing swaps in resource allocation systems and we thus uniformly transfer notions from the concurrent setting of games to resource allocation systems as well. Proceeding further, we get a similar notion for voting systems, where again a subset of voters can get together and agree on revising their expressed preferences, thus leading us to coalitional improvement paths and coalitional FIP.

In general, we might want to specify reachability of a set of distinguished nodes satisfying some property. For instance, in the context of allocation systems, we are interested in nodes that are envy free. An agent envies an agent at node if there exists a node such that and the allocation for at is the same as the allocation for at . A node is envy-free if no player envies another at that node. We might then want to assert that an envy free node is reachable from any node. Note that we only need to enrich the first order vocabulary to speak of etc to express envy-freeness, and the lfp operator is sufficient to specify reachability of such nodes.

In the context of voting systems, a similar strengthening of the syntax of atomic formulas can lead to amusing specification of reachability of singular profiles in which all voters have identical first and last preference but disagree on all candidates ranked by them in between.

Further, the counting quantifier can give us interesting relaxations of improvement dynamics. For instance consider the following specifications:

This specifies that at most nodes have finite improvement paths originating from them. Clearly, such specifications are of greater interest in voting systems than in the others. Here the lfp operator is in the scope of the counting quantifier. In the following specification, we have them the other way about.

Consider the housing market model mentioned in the previous section. This framework corresponds to a resource allocation problem where a valid allocation corresponds to a bijection between agents and resources and each agent is initially assigned an item. Stability is an important solution concept in this setting. The dynamics in which agents start with the initial allocation and repeatedly exchange items provided it is a profitable move for all the agents involved in the exchange is quite natural. The obvious question is whether this process converges to a stable outcome. If we model the associated improvement dynamics graph as exchange between pairs of agents then each action corresponds to resolving a blocking pair. The sentence sink then refers to the existence of 2-stable outcomes and acyclic asserts that every sequence of blocking pair leads to a 2-stable outcome. If we interpret the improvement dynamics graph as capturing exchange of items within a coalition of agents then these sentences correspond to existence and convergence to core-stable outcome. For the housing allocation problem, the existence of core-stable outcome is guaranteed by the Top Trading Cycle algorithm. There are variants of the housing allocation problem where stable outcomes are not always guaranteed to exists (for instance, in the presence of externalities). In this case, by using the counting quantifier, we can make interesting assertions like the number of nodes that have finite improvement paths originating from them.

4 Model Checking Algorithm

In this section we discuss the model checking problem for MLFPC. Given and a -structure , the model checking problem is to check if . We show that the logic admits an efficient model checking procedure (Algorithm 1). The main idea is to use the model checking algorithm for FO and modify it accordingly for the newly introduced quantifiers of the count and the least fixed point.

[width=5.5in,breakable, title= Algorithm 1 for Checking of ] Input :
Output : Sol
= subformulas of /* ordered via subformula ordering */
Q =

  1. for

    1. switch type of

      1. case is an atomic formula
        is computed by reading off the structure

      2. case
        is just the negation of /*since is already computed */

      3. case
        Let P = , Q = , R =
        for

        1. if
             

        2. else

      4. case
        for
           if
            

      5. case
        Let count = 0
        for

        1. for
              if then count = count + 1

        2. if count k then

        3. count = 0

      6. case
        lfp() /*subroutine call */

[width=5.5in, breakable, title= Algorithm 2 : lfp() - Subroutine call for computing the least fixed point ] Input :
Output :
Let ,
for

  1. iter = ,

  2. do

    1. iter =

    2. /* call to a FO-model checking procedure, where the second order variables are fixed */

    3. for
          if then

  3. while

Theorem 1.

Given and , Algorithm 1 decides if in polynomial time.

Proof.

Given , let denote the set of subformulas of and denote the corresponding subformula ordering. The idea follows the algorithms outlined for model checking procedures in FO and combines with what is known about the least fixed point. We maintain a polynomial time computable relational list of polynomial size. We will basically follow the proof of the first order logic and argue similarly for the count and least fixed point operators introduced. Let be the underlying domain of . For each let where if and otherwise.

Induction Hypothesis.

For all , is of size and can be computed in time .

Base Case.

When is an atomic formula, then can be directly computed from . Since each of the relations defined in are polynomial in the size of A and it would take a linear pass across the relations expressed to compute . Thus, we can conclude that the total time taken is also polynomial in the size of A.

Induction Case.

  • By, I.H, since , we would have already computed . which can be computed by an algorithm running in time .

  • By, I.H. we have in polynomial time been able to maintain the polynomial sized lists and . Then the procedure outlined in the algorithm would take time and maintains a list of similar size.

  • By, I.H. we have in polynomial time been able to maintain the polynomial sized lists . The algorithm outlined would take which would also happen to be the size of the list maintained by .

  • By, I.H. we have in polynomial time been able to maintain the polynomial sized lists . The outlined procedure computes in time .

  • First we note, , and . What essentially happens here is that the second order variable gets used to generate an inductive relation via the fixed point computation and the newly introduced first order variable u is utilised to check validity of these formulas. So, unfortunately we cannot directly use to give a polynomial time procedure to generate . We would rather use the fact that the least fixed point computation is a polynomial time computation even in the case of the first order logic with the count introduced.

    If we look at the inductive procedure to compute the fixed point we see that the monadic fixed point operator starts at an empty set and then converges to a subset of A. And at each stage there is an increment in the number of elements of the output set by at least 1. Therefore this will run in maximum time. Now for the call to model checking at each stage we notice that since the formulas get fixed values assigned at each model checking call, we actually operate a FO-model checking call, which we know happen to take time. We have a precomputation of , which we can additionally make use of to reduce the time in the following manner.
    For each choice of the entire instance gets fixed and in thus a linear pass through we would get to know whether the choice of satisfies the formula or not. Therefore each stage that does the FO-model checking would take time at most . Therefore total time taken by the entire lfp computation part is . We are interested in the data complexity of our procedure, where we can ignore the size of the formula, which typically happens to be of lesser size than the model over which the model checking procedure is held (reflects the practical circumstances). So we can conclude that our procedure runs in time .

Complexity of Algorithm 1. The algorithm iterates over all subformulas in increasing order. The worst case running time of the lfp procedure for inputs and is , where . Thus the running time of Algorithm 1 is . For the specific properties mentioned in section 3.3, note that the corresponding MLFPC formulas refers to one second order variable and two first order variables. Thus for all the properties mentioned in section 3.3, the model checking procedure runs in time .

In the context of improvement graphs, if there are agents and at most choices for each agent, the size of the associated improvement graph is . Since it is possible to have a compact representation for certain subclasses of strategic form games, for instance, polymatrix games [25], the size of the improvement graph structure can be exponential in the representation of the game. Thus the model checking procedure, while polynomial on the size of the underlying improvement graph, can in principle, be exponential in the size of the game representation. This observation may not be very surprising since even for restricted classes of games like 0/1 polymatrix games, checking for the existence of Nash equilibrium is known to be NP-complete [5].

5 Discussion

We see this paper as a preliminary investigation, hopefully leading to a descriptive complexity theoretic study of fundamental notions in games and interaction. It is clear that fixed-point computations underlie the reasoning in a wide variety of such contexts, and logics with least fixed-point operators are natural vehicles of such reasoning. We expect that this is a minimal language for improvement dynamics, but with further vocabulary restrictions that need to be worked out. Proceeding further, we would like to delineate bounds on the use of logical resources for game theoretic reasoning. For instance, one natural question is the characterization of equilibrium dynamics definable with at most one second order (fixed-point) variable.

Expressiveness needs to be sharpened from the perspective of models as well. We would like to characterize the class of improvement graphs for different subclasses of games, resource allocation systems and voting rues, considering the wide variety of details in the literature. This would in general necessaite enriching the logical language and we wish to consider minimal extensions.

Another important issue is the identification of subclasses that avoid the navigation of huge improvement graphs. Potential games provide an interesting subclass and they correspond to some appropriate allocation rules and forms of voting (under specific election rules). But these are only specific exemplifying instances, studying the stucture of formulas and their models will (hopefully) lead us to many such correspondences.

An important direction is the study of infinite strategy spaces. Clearly the model checking algorithm needs a finite presentation of the input but this is possible and it is then interesting to explore convergence of fixed-point iterations.


Acknowledgements. We thank the reviewers for their insightful comments. We thank Anup Basil Mathew for discussions on improvement dynamics. Sunil Simon was partially supported by grant MTR/ 2018/ 001244.

References

  • [1]
  • [2] R. Alur, T. A. Henzinger & O. Kupferman (2002): Alternating-Time Temporal Logic. Journal of the ACM 49, pp. 672–713, doi:http://dx.doi.org/10.1145/585265.585270.
  • [3] Krzysztof R Apt, Bart de Keijzer, Mona Rahn, Guido Schäfer & Sunil Simon (2017): Coordination games on graphs. International Journal of Game Theory 46(3), p. 851–877, doi:http://dx.doi.org/10.1007/s00182-016-0560-8.
  • [4] Krzysztof R Apt & Sunil Simon (2015): A classification of weakly acyclic games. Theory and Decision 78(4), pp. 501–524, doi:http://dx.doi.org/10.1007/978-3-642-33996-7˙1.
  • [5] Krzysztof R. Apt, Sunil Simon & Dominik Wojtczak (2015): Coordination Games on Directed Graphs. In: Proc. of 15th International Conference on Theoretical Aspects of Rationality and Knowledge, doi:http://dx.doi.org/10.4204/EPTCS.215.6.
  • [6] J. van Benthem (2001): Games in dynamic epistemic logic. Bulletin of Economic Research 53(4), pp. 219–248, doi:http://dx.doi.org/10.1111/1467-8586.00133.
  • [7] J. van Benthem (2002): Extensive games as process models. Journal of Logic Language and Information 11, pp. 289–313, doi:http://dx.doi.org/10.1023/A:1015534111901.
  • [8] Johan van Benthem & Amélie Gheerbrant (2010): Game Solution, Epistemic Dynamics and Fixed-Point Logics. Fundamenta Informaticae 100(1-4), pp. 19–41, doi:http://dx.doi.org/10.3233/FI-2010-261.
  • [9] G. Bonanno (2001): Branching Time Logic, Perfect Information Games and Backward Induction. Games and Economic Behavior 36(1), pp. 57–73, doi:http://dx.doi.org/10.1006/game.1999.0812.
  • [10] Sylvain Bouveret, Yann Chevaleyre & Nicolas Maudet (2016): Fair Allocation of Indivisible Goods, chapter 12. Handbook of Computational Social Choice, Cambridge University Press.
  • [11] Simina Branzei, Ariel D. Procaccia & Jie Zhang (2013): Externalities in cake cutting. In: Proceedings of the 23rd IJCAI, pp. 55–61.
  • [12] E. Budish (2011): The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy 119(6), pp. 1061–1103, doi:http://dx.doi.org/10.1086/664613.
  • [13] Krishnendu Chatterjee, Thomas A. Henzinger & Nir Piterman (2010): Strategy logic. Information and Computation 208(6), pp. 677–693, doi:http://dx.doi.org/10.1016/j.ic.2009.07.004.
  • [14] Y. Chevaleyre, U. Endriss & N. Maudet (2017): Distributed Fair Allocation of Indivisible Goods. Artificial Intelligence 242, pp. 1–22, doi:http://dx.doi.org/10.1016/j.artint.2016.09.005.
  • [15] Anastasia Damamme, Aurélie Beynier, Yann Chevaleyre & Nicolas Maudet (2015): The power of swap deals in distributed resource allocation. In: Proceedings of 14th International Conference on Autonomous Agents and Multiagent Systems, pp. 625–633.
  • [16] H. Ebbinghaus & J. Flum (1999): Finite Model Theory. Springer, doi:http://dx.doi.org/10.1007/3-540-28788-4.
  • [17] R. Engelberg & M. Schapira (2011): Weakly-Acyclic (Internet) Routing Games. In: Proc. 4th International Symposium on Algorithmic Game Theory (SAGT11), Lecture Notes in Computer Science 6982, Springer, pp. 290–301, doi:http://dx.doi.org/10.1007/978-3-642-24829-0˙26.
  • [18] Etsushi Fujita, Julien Lesca, Akihisa Sonoda, Taiki Todo & Makoto Yokoo (2015): A Complexity Approach for Core-Selecting Exchange with Multiple Indivisible Goods under Lexicographic Preferences. In: Proceedings of the 29th AAAI Conference on Artificial Intelligence, pp. 907–913, doi:http://dx.doi.org/10.1613/jair.1.11254.
  • [19] M. Ghodsi, H. Saleh & M. Seddighin (2018): Fair Allocation of Indivisible Items With Externalities. CoRR abs/1805.06191. Available at http://arxiv.org/abs/1805.06191.
  • [20] V. Goranko (2003): The Basic Algebra of Game Equivalences. Studia Logica 75(2), pp. 221–238, doi:http://dx.doi.org/10.1023/A:1027311011342.
  • [21] Laurent Gourves, Julien Lesca & Anaëlle Wilczynski (2017): Object allocation via swaps along a social network. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI’17), pp. 213–219, doi:http://dx.doi.org/10.24963/ijcai.2017/31.
  • [22] Martin Grohe (1994): The Structure of Fixed Point Logics. Albert-Ludwigs-Universitat.
  • [23] P. Harrenstein, W. van der Hoek, J.J. Meyer & C. Witteven (2003): A Modal Characterization of Nash Equilibrium. Fundamenta Informaticae 57(2-4), pp. 281–321.
  • [24] W. van der Hoek, W. Jamroga & M. Wooldridge (2005): A Logic for Strategic Reasoning. Proceedings of the Fourth International Joint Conference on Autonomous Agents and Multi-Agent Systems, pp. 157–164, doi:http://dx.doi.org/10.1145/1082473.1082497.
  • [25] E.B. Janovskaya (1968): Equilibrium Points in Polymatrix Games. Litovskii Matematicheskii Sbornik 8, pp. 381–384.
  • [26] Bernd Kawald & Pascal Lenzner (2013): On Dynamics in Selfish Network Creation. In: Proceedings of the 25th ACM Symposium on Parallelism in Algorithms and Architectures, ACM, pp. 83 – 92, doi:http://dx.doi.org/10.1145/2486159.2486185.
  • [27] O. Lev & J. S. Rosenschei (2012): Convergence of iterative voting. In: Proceedings of AAMAS-2012, pp. 611–618.
  • [28] Leonid Libkin (2013): Elements of finite model theory. Springer, doi:http://dx.doi.org/10.1007/978-3-662-07003-1.
  • [29] J.R. Marden, G. Arslan & J.S. Shamma (2007): Regret based dynamics: convergence in weakly acyclic games. In: Proceedings of the Sixth International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2007), IFAAMAS, pp. 194–201, doi:http://dx.doi.org/10.1145/1329125.1329175.
  • [30] Reshef Meir, Maria Polukarov, Jeffrey S.Rosenschein & Nicholas R.Jennings (2017): Iterative voting and acyclic games. Artificial Intelligence 252, pp. 100–122, doi:http://dx.doi.org/10.1016/j.artint.2017.08.002.
  • [31] I. Milchtaich (1996): Congestion Games with Player-Specific Payoff Functions. Games and Economic Behaviour 13, pp. 111–124, doi:http://dx.doi.org/10.1006/game.1996.0027.
  • [32] D. Monderer & L. S. Shapley (1996): Potential Games. Games and Economic Behaviour 14, pp. 124–143, doi:http://dx.doi.org/10.1006/game.1996.0044.
  • [33] Roger B. Myerson & Robert J. Weber (1993): A Theory of Voting Equilibria. The American Political Science Review 87(1), pp. 102–114, doi:http://dx.doi.org/10.2307/2938959.
  • [34] R. Parikh (1985): The logic of games and its applications. Annals of Discrete Mathematics 24, pp. 111–140, doi:http://dx.doi.org/10.1016/S0304-0208(08)73078-0.
  • [35] R. Ramanujam & S. Simon (2008): Dynamic logic on games with structured strategies. In: Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR-08), AAAI Press, pp. 49–58.
  • [36] A. Reijngoud & U. Endriss (2012): Voter response to iterated poll information. In: Proceedings of AAMAS-2012, pp. 635–644.
  • [37] R. W. Rosenthal (1973): A Class of Games Possessing Pure-Strategy Nash Equilibria. International Journal of Game Theory 2(1), pp. 65–67, doi:http://dx.doi.org/10.1007/BF01737559.
  • [38] Nicole Schweikardt (2005): Arithmetic, first-order logic, and counting quantifiers. ACM Transactions on Computational Logic 6(3), pp. 634–671, doi:http://dx.doi.org/10.1145/1071596.1071602.
  • [39] Nicole Schweikardt (2006): On the expressive power of monadic least fixed point logic. Theoretical Computer Science 350, p. 325–344, doi:http://dx.doi.org/10.1016/j.tcs.2005.10.025.
  • [40] L. S. Shapley & H. Scarf (1974): On cores and indivisibility. Journal of Mathematical Economics 1(1), pp. 23–37, doi:http://dx.doi.org/10.1016/0304-4068(74)90033-0.
  • [41] Zhaohong Sun, Hideaki Hata, Taiki Todo & Makoto Yokoo (2015): Exchange of Indivisible Objects with Asymmetry. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence, pp. 97–103.
  • [42] D. Walther, W. van der Hoek & M. Wooldridge (2007): Alternating-time Temporal Logic with Explicit Strategies. In: Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge (TARK-2007), pp. 269–278, doi:http://dx.doi.org/10.1145/1324249.1324285.
  • [43] H. Peyton Young (1993): The evolution of conventions. Econometrica 61(1), pp. 57–84, doi:http://dx.doi.org/10.2307/2951778.