Intention Games

Intention Games

Aditya Ahuja Department of Computer Science and Engineering
Indian Institute of Technology Delhi
aditya.ahuja@cse.iitd.ac.in
Abstract

Strategic interactions between competitive entities are generally considered from the perspective of complete revelations of benefits achieved from those interactions, in the form of public payoff functions in the announced games. In this work, we propose a formal framework for a competitive ecosystem where each player is permitted to deviate from publicly optimal strategies under certain private payoffs greater than public payoffs, given that these deviations have certain acceptable bounds as agreed by all players. We call this game theoretic construction an Intention Game. We formally define an Intention Game, and notions of equilibria that exist in such deviant interactions. We give an example of a Cournot competition in a partially honest setting. We compare Intention Games with conventional strategic form games. Finally, we give a cryptographic use of Intention Games and a dual interpretation of this novel framework.

1 Introduction

Game theory [1] considers non-cooperative strategic interactions among players with public knowledge of the actions available and payoff structures for the involved parties. Even in the case of games with incomplete information, the framework of the game is consistent with those of complete information games. This framework consists of a direct, disclosed payoff for each player as a function of the actions chosen by all players. These actions are chosen by players individually with selfish interests alone. It is this notion of rationality that dictates strategic choices and principles of equilibria.

However, rational entities my compete amongst themselves with a partial disclosure of payoffs achieved out of the competition. As long as each competing entity is in the knowledge that other participating entities might have payoffs over and above the disclosed payoffs, which is true for the said entity also, it is reasonable to assume applicability of novel game structures in this partially honest setting. Further, game equilibria should dictate that there don’t exist players which are being dominantly unfair to other players through excessive deviations under the disclosed ‘public game’ agreed upon by all (potentially dishonest) players.

In order to address this problem of partially honest competition, we propose a new game theoretic framework called Intention Games. Our framework permits strategic games among partially honest players with publicly declared payoffs less than or equal to actual payoffs. Moreover, these hidden payoffs per player might change per iteration of the Intention Game and can potentially result from secret contracts of each player with hidden parties. We define best responses in this (potentially devious) ecosystem and equilibria as a function of the publicly observed deviations in actions and payoffs. We give an example of a Cournot Competition (Section 3.1, [1]) in a partially honest setting. We give a use of Intention Games for the discovery of a cryptographic key. We also give a possible dual interpretation of an Intention Game.

This paper is organized as follows. Firstly, we motivate the need of a new game paradigm in a dishonest setting in Section 2. In Section 3 we formally and rigorously define an Intention Game, and it’s equilibria. We give an example of an Intention Game through an Intention Cournot Duopoly in Section 4. In Section 5, we contrast the Intention Game framework with conventional strategic form games. We give a cryptographic key discovery game and the dual interpretation of Intention Games in Section 6. We compare Intention Games with existing game frameworks in Section 7. We close the paper in Section 8 with concluding arguments and future directions for this work.

2 Motivating a New Game Paradigm

As claimed before, classical game theory consists of complete declarations of possible strategies of each player and the payoffs associated with the actions taken collectively by the players involved. However, there exists a simple, logical extension of this competitive ecosystem where the collective actions taken in their self interest by the competing players have implications of higher payoffs due to hidden parties secretly colluding with each player in isolation. These private increments in payoffs as a function of action profiles chosen under a certain notion of best response in the public part of the game result in a cumulative private payoff greater than, or equal to, the public payoff (payoff announced in the public part of the game). We call this extension of conventional games as our Intention Games framework.

As a first example, consider a conventional game in a war where the players are individual sovereign nations who are bound by a treaty, say , the actions are armed troop deployments in a particular geographic region at a certain time, and payoffs are functions giving a numerical representation of the victory in the battle. Now given an action profile for all countries following , the can be used as a certificate by each country as a bargaining chip for troop deployment for alternate treaty/treaties between and other countries under . Note that is independent of and consequently the resulting cumulative payoff for is private with respect to and above that achieved in .

As a second example, consider HTTP contracts between Content Delivery Networks (CDNs) and a single Internet Service Provider (ISP) [6], as our game . In this case the action set per CDN are HTTP objects sourced by that single ISP from that CDN. The payoffs are revenues generated from the HTTP applications served by the ISP’s customers. Here again, an action profile of HTTP objects delivered could allow each CDN in to negotiate with other ISPs it has contracts with, resulting in a higher private payoff. Essentially, would be a certificate for the CDN of availability / service record of the HTTP objects it controls.

Note that through our examples, we wish to reiterate that Intention Games is a framework of mutual acceptance of dishonest behaviour among involved players. A dishonest strategy for an optimal private payoff by one player might result in suboptimal payoffs for the other honestly participating players. Therefore, this dishonesty must have bounds as the game evolves. We formally capture this notion though our equilibria in Sections 3.3, 3.4.

3 Intention Games: Formal Definition

3.1 Notation and Preliminaries

Let denote the set of all probability distributions on universe . We will use the set notation . For brevity, we will denote the split of a vector on an index as while implicitly preserving the order of elements. The denotes all indices except .
Note that we will only give outlines of equilibria computation. The complexity arguments are implied from classical game theory, with Nash Equilibrium being PPAD-Complete [2].

3.2 Definitions

We first give the formal definition of an Intention Game.

Definition 1 (Intention Game)

An Intention Game is a repeated game given by a tuple where

  1. is the set of players.

  2. is the set of actions available to player . Also is the set of all action profiles.

  3. Each player has a public payoff function and a private payoff function which can change per iteration .
    It is also the case that for all iterations, .

  4. All players collectively agree on a public strategic form game
    , which is the ‘public image’ of the Intention Game.

  5. Each player strategises according to it’s ‘self reflection’ of the Intention Game: .

  6. For each player , there exists a partition of such that
    and .

Note that in point under Definition 1 above, we extend the notion of a partition to permit to be empty. Also, intuitively, is the set of publicly deviant action profiles for player , with being a witness of deviation for profile .

Definition 2 (-Intention Game)

A given Intention Game is a -Intention Game if in each iteration of the game there exist at most players such that and for the remaining players it is the case that .

For each iteration of the Intention Game, we call the set of deviant players, and as the set of non-deviant players. It’s an easy verification that and .
For the rest of the paper, we will consider only -Intention Games for the notion of our best response strategies and equilibria.

We now give how best responses are defined in the Intention Games framework. Note that these are just reinterpretations and extensions of the Nash equilibrium (Chapter 2, [1]).

Definition 3 (Best Response Set)

Given a strategy universe , a payoff , and a complementary strategy profile , the Best Response Set is given by
.

Definition 4 (Best Response Profiles)

Given the ‘public image’ and each ‘self reflection’ of the Intention Game, the best response profiles are given by

Theorem 3.1 (Best Response Profile Dependencies)

Given an action profile , if for some , , then

  1. .

  2. .

Proof

Since for each player , the best response choices between and only differ in the payoff of player , our proof will only consider choices as a function of and .
Proving . Let’s say . Then there exists a witness of deviation such that . So is not a best response under payoff given the complementary action profile . Thus and .
Proving . Let’s say is not a member of the best response set under payoff given the complementary action profile . Then there exists an such that . Also since player is playing best responses under payoff given complementary action profile it is true that (otherwise would have chosen the action ). The last two lines imply .
Proving . This statement is the equivalence complement (for propositions and , if and only if ) of statement , which has been proved.

We also give the implication of the dishonest player’s actions on the honest players.

Corollary 1 (Fallout for Honest Players)

Given for some dishonest player , and , then is a suboptimal payoff action profile for all honest players as .

Note that in corollary 1 as , the deviation witness that maximizes corresponds to the Nash optimal strategy for under .

3.3 A Repeated Pure Strategy Equilibrium

We first give a notion of equilibrium which is captures how many instances of publicly observed deviations under are seen by all players upto the current run of the Intention Game.

Definition 5 (Honesty Equilibrium)

A pure-strategy profile vector is a -Honesty Equilibrium if after iterations of the Intention Game,
given that ,
it is the case that .

Computation.

We assume that the computation of a pure-strategy Nash equilibrium for is a given. We give the method for computing the Honesty equilibrium, as an invariant under . Let’s say is the Honesty equilibrium bound upto epoch . Now given , compute public deviation, using theorem 3.1, by testing membership of in for each . Note that since each player is playing best responses under it’s private payoff, this membership can be tested by only finding an such that . If there exists a single player for which is publicly deviant, set . Otherwise set .

3.4 A Mixed Strategy Equilibrium

We now give a mixed-strategy equilibrium for a -Intention Game where the deviant player persists with his higher payoff for polynomially many rounds in . For each of those rounds, players are non-deviant.

Definition 6 (Deviation Equilibrium)

A mixed-strategy profile vector
, where is a mixed-strategy best response under , is a -Deviation Equilibrium if under .

Computation.

We assume that the computation of a mixed-strategy equilibrium for is a given: first we compute the mixed-strategy Nash equilibrium under and then replace the th player’s (randomized) Nash optimal with the randomized best response under (by keeping the Nash optimal constant for all ).
We first give the method by which a player can compute his own deviation bound . Given the distribution computed in the previous step, it is straightforward to compute the distribution , if the functions are deterministic and efficient (polynomial time computable). So, we can determine as the expected value of .
We now give the method whereby a player can compute a lower bound for given a sufficiently long stream of realizations of . For an arbitrary iteration of the Intention Game, let be the realization of the strategies played by all players. If , find an such that is maximized. If , . By the law of large numbers, .
Finally, .

Discussion.

It is clear from the definition of both the Honesty and Deviation equilibria, the game is more fair as long as and are small. So these equilibria definitions can be used by each player to announce the terms of competition. For instance, players might agree on a -Honesty equilibrium conforming game as long as , for some contractual constant . As another case, players might agree on a -Deviation equilibrium conforming game as long as for some previously announced constant . Whenever are exceeded, players terminate the Intention Game.

3.5 Equilibrium Degeneration: Uncaught Deviation under certain Private Payoffs

We now discuss functional relations where best responses under private payoffs are also best responses under public payoffs, even in the case that .
Consider the function family . It is an easy verification that in such cases, and the deviant (cheating) player would never be caught and computing the Nash equilibria for would suffice.

4 An Example: Intention Cournot Duopoly

We now give a concrete example of an Intention Game. We extend a conventional Cournot duopoly where two firms compete in the supply of a single homogenous product to a single market with identical cost functions and symmetric payoffs. Our extended ‘Intention Cournot Duopoly’ (ICD) involves a secret contract with a mobile but hidden second market which is in contact with at most one firm at any point in time. Both firms have a symmetric secret contract with this hidden mobile market which defines a (higher than public payoff) private payoff in the Intention Game whenever the corresponding firm participates in the contract in the event of a contact with the hidden market (see Figure ).

Market M

Hidden Market H

Figure 1: An Intention Cournot Duopoly between Firms and , given a secret contract with a Hidden Market.

More formally, given a market M and a hidden mobile market H with a secret contract , is an Intention Cournot Duopoly where

  1. ,   .

  2. If H is in contact with firm , it will take the same supply as to M, compensate production costs, and pay half of the supply from to it. Otherwise no trade.    ()

  3. ,   (payoff from M).

  4. where
    or under contract ,   .

Note that the price and cost functions are expanded inline in point . Also is a single player .

4.1 Honesty and Deviation Equilibria

Using superscripts and for best responses under public and private payoffs respectively, it is an easy calculation that the best response profile sets are singletons:
and
whenever is applicable and succeeds.

Note that is a (maximal) deviation witness of .

According to our statement of the ICD, at any iteration of , can succeed for at most one , so is a -Intention Game. Thus, the factor of an Honesty Equilibrium can increment by in an iteration of the ICD if for the said iteration succeeds and consequently .

Also . So given best responses under and , there exists a -Deviation Equilibrium.

Finally, note that a deviation by player costs player : .

5 Comparison with Conventional Strategic Form Games

There is an instance when an Intention Game is identical to a conventional (underlying) strategic form game. Consider the case where for all iterations of the Intention Game , , for all players . In this case, for all iterations of , . Further, the Nash equilibrium will hold per iteration of and any evolution of the Intention Game would result in -Honesty and -Deviation equilibria. This can be intuitively be seen from the fact that for all players in any evolution of the Intention Game, the set of publicly deviant action profiles will always be empty.

There is one more instance where the best responses for an Intention Game are identical to those of the underlying strategic form game, even when the two games are different (we have also discussed this in Section 3.5). We give the function family for the Intention Game . Now it is an easy verification that although , we have . Here again, the Nash equilibrium will hold per iteration of and any evolution of the Intention Game would result in -Honesty and -Deviation equilibria.

6 Possible Uses and Counter Interpretations

In this section, we give an instance of how an Intention Game can be leveraged to realize a cryptographic task for a competitive community. Following this, we give a counterintuitive but relevant dual of the Intention Game, which we call a Selfless Game.

6.1 A Cryptographic Key Discovery Game

We first give an Intention Game where players interested in discovering a cryptographic key deviate from publicly optimal strategies in the event of assurance of obtaining the key at the end of the game. In this case, the deviation is like an announcement that the player has achieved the goal of participating in the game, the goal being to obtain the key.

Assuming a familiarity with cryptography [7], let be the security parameter. Let poly(), exp() and negl() be the set of all polynomial, exponential and negligible functions in . We will stretch notation by sometimes placing these function classes in place of functions that belong to these classes. Let the length of the target key be . Let the ‘key discovery table’ be . is a table with poly() entries of bits each. There is a hidden negotiator who visits at most one player in each iteration (constituting a -Intention Game). A player successfully gets if the strategy profile in the current iteration of the Intention Game is a member of and the hidden negotiator visits in the current iteration.

We formally state the Key Discovery Intention Game . We assume there is a hidden negotiator H with a secret contract with all players. The negotiator chooses an arbitrary permutation of to visit one player in each iteration of . The strategies, contract and payoffs are given by ( denotes uniform distribution):

  1. ,
    Public , .

  2. If H is in contact with player , it will check if . If membership succeeds, gets a payoff of unit. Otherwise payoff.    ()

  3. , if , otherwise .

  4. In iteration , where
    or under contract as applicable in iteration ,   .

Discussion.

Strategies for each player are samples from . Concatenated strategies are tested for membership in . Given the size of , the game is guaranteed to end in polynomial time. If player strategies are coming from the uniform distribution on , the probability that any member of is seen is negl(). This permits an announcement of key discovery to come from . So if succeeds in iteration , comes from . Thus, is a publicly deviant action profile with deviation witness as any member of . Note that we have made a slight abuse in the definition of by using an action profile from a previous iteration. However, the ‘feedback’ is legitimate and acceptable due to the nature of the negotiator’s visits to players ( if , then cannot succeed in iteration ). Note that is small.
Finally, for a requirement that players discover the key, the Intention Game can run for the smallest till a -Honesty Equilibrium is achieved (we can do a minimal derandomization to claim pure strategies).

6.2 Selfless Games: Private Payoff less than Public Payoff

Now, we give a dual framework of an Intention Game . A ‘Selfless Game’ is a dual of in the sense that for all iterations of , the private payoff is less than the public payoff: .

Note that this definition of a game where best responses don’t maximize payoffs would seem counterintuitive but still is relevant: such incetivizations are possible and rooted in behavioural psychology. For instance, consider a game between a mother and a son in which best responses coming from the mother are suboptimal under . However, the ‘action profile certificates’ might incentivize the mother to get an alternative reward (for the mother) from the father, compensating the sub optimality achieved in . Thus although the mother does not win against the son, she ultimately wins due to a ‘contract’ with the father.

Having suggested a possible dual of Intention Games, we conclude that a rigorous motivation and definition of Selfless Games is not in the scope of our current work.

7 Related Work

Classical game theory considers extensively models of competition in the form of games with incomplete information [1]. Bayesian games (Chapter 9, [1]) allow players to have imperfect information about some aspect of other players, but have beliefs about those aspects through some probability distribution. Stochastic games [5] are extensive form games where the transitions taken by players are random variables. In both these cases, the game theoretic model is of incomplete information, but can be modelled as a distribution on the uncertainty. However, Intention Games do not permit any freedom to model / analyze structure of the private payoffs, which can change arbitrarily per iteration of the game.

Secondly, there do exist studies on learning in games [4]. Again, methods such as fictitious play are inapplicable since the private payoff can change per iteration of the game. Also no player can learn the exact private payoff of any other player, only the lower bound on the payoff deviation of each player through the public information given by all players during the game.

Traditional game theory considers rational choices and utility maximization as a norm. However, there have been models for deviation from rational behaviour owing to beliefs, social issues, group issues under the formalism of behavioural game theory [3]. Even so, the deviations considered in behavioural game theory occur from eccentricities implicit in the participating players due to external effects. In comparison, we do not compromise on the notion of rationality in defining Intention Games. While considering players to be rational, but partially honest, we propose our competitive framework.

8 Conclusions and Future Work

In this work, we have proposed a novel framework for dishonest cooperation among competitive entities where participating players can deviate from publicly optimal strategies under certain predefined contractual bounds. Our novel game theoretic construction called Intention Games is an extension of conventional strategic games but allows the flexibility of simultaneous dishonesty among players. We have given an example of an Intention Cournot Duopoly to demonstrate the implications of publicly devious behaviour on outcomes and equilibria. We have also demonstrated how Intention Games can be used for the task of competitively obtaining a cryptographic key. Finally, we have outlined a dual framework of Intention Games, which we call Selfless Games.

In future, we would like to explore equilibria for -Intention Games with . We would like to give a similar ‘Intention’ framework to dynamic games (Chapter 5, [1]). We would also like to explore a rigorous motivation and definition of Selfless Games. Finally, we would like to build cryptographic primitives with underlying hardness coming from an Intention Game framework.

Acknowledgements.

I would like to thank Dr. Ranjan Pal (Research Scientist, University of Southern California) and Prof. Vinay J. Ribeiro (Associate Professor, IIT Delhi) for their constant support.

References

  • [1] Osborne, Martin J. “An Introduction to Game Theory.” Vol. 3. No. 3. New York: Oxford university press, 2004.
  • [2] Roughgarden, Tim. “Algorithmic Game Theory.” Communications of the ACM 53.7 (2010): 78-86.
  • [3] Camerer, Colin F. “Behavioral Game Theory: Experiments in Strategic Interaction.” Princeton University Press, 2011.
  • [4] Fudenberg, Drew, and David Levine. “Learning in Games.” European economic review 42.3 (1998): 631-639.
  • [5] Shapley, Lloyd S. “Stochastic Games.” Proceedings of the national academy of sciences 39.10 (1953): 1095-1100.
  • [6] Kurose, James F. “Computer Networking: A top-down approach featuring the Internet”, 3/E. Pearson Education India, 2005.
  • [7] Katz, Jonathan, and Yehuda Lindell. “Introduction to Modern Cryptography”. CRC press, 2014.
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