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Inception Games: Foundation and Concepts


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Mathematical generalization of dynamic, evolutional, and infinitely recursive multi-agent social conflict games loosely based on some concepts from Christopher Nolan's screenplay/movie Inceptions.

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Inception Games: Foundation and Concepts

  1. 1. Inception Games: Foundation and Concepts Alfredo Sepulveda Colorado Technical University CS-EM September 19, 2013 In partial fulfillment of post-doctoral research requirements ©2013 Copyright, all rights reserved by Alfred Sepulveda for all slides
  2. 2. Agenda • Introduction: Inceptions • Why Inception Models (as Recursions) Are Important • Generalizations to the Movie Scenario • Components of Generalized Inception Games • Decision Branching and Game History Trees • Description of Inceptions As Strategic Recursive Games • Recursion in Inceptions • Some Research Questions • Theories, Concepts, and Hypotheses • Limitations • Research Methodologies • Data Collection and Analysis • Inception Payoffs • Belief Revision of Agent Strategies Inception Games
  3. 3. Agenda (con’t) Inception Games • Conscious Awareness and Social Order • Validating Beliefs and Pre-comprehension: Spinoza to Descartes • Measuring Epistemic Belief Revision Potential • Making for Better Decision Makers and Judges of Truth Values • Expanding and Expressing Risk in Games and Decision Making • Naturally Sensing Risk: Virtual Hair on the Back of Your Neck • What Does Risk Look Like? • Inceptions as Recursive Automata • Computation of Game Solutions • Belief Regions, Solutions, and Equilibria • Features of Inceptions • Supporting Content for Inception Game Concepts • Future Considerations • Conclusions
  4. 4. Loosely based on 2010 screenplay/movie by Christopher Nolan Premise: groups of individuals simultaneously enter “dream world” scenarios involving a person of interest in order to coalesce information from that individual through befriending, trust building, influence, and coercion. Consecutively potential embedded dream (inception) levels may be entered such that each descended level nearly suspends current time for current level in order to achieve a possible inception and essentially “expand perceived time” by a certain factor: level no. perceived time epoch 0 (reality) 1m 1 (first dream) 12m 2 (second dream) ~410m* 3 (third dream) ~8,637m* 4 (limbo) ~182,008m* …… …… ? *based on nonlinear exponential model of 10hr flight, 1 week first dream, 6 month second dream, and 10 years third dream Introduction: Inception Games Inception Games
  5. 5. Why Inception Models (as Recursions) are Important • Anthropomorphic thought and hence decision-making is recursively structured possibly of an emergent fractal nature (if you believe Kurzweil and cognitive scientists starting from the 1980s). • Recursion in automata and computation is a powerful metaphor and mechanism for GAI. • Visualizing general risk in actions can be thought of as a recursive emergent phenomena when considering macro, meso, and micro views of decisions over an epoch. • Recursions are natural models for conflicts – every thought process is a type of conflict. • Inceptions introduce the potential for ubiquitous and continuously connected human- machine coalitions for decision-making – singularity types of transhuman behavior (another Kurzweilian fantasy). • Inceptions may generalize the concepts behind conflict gaming. • Inceptions may provide a model for investigating belief revision dynamics Inception Games
  6. 6. Generalizations to the Movie Scenario • Multiple coalitions with mixed general inter-coalition agendas and linkage • Multiple n-agencies and belief regress • Quantum-gravity (QG) causal models for recursive temporal inception levels • Massively multiple agent coalitions • Extension of psychological affects • Payoffs are tied to inception information recovered and time-discounted inception levels • Infinite version of inception levels (hyper-reality and hyper-limbo) • General uncertainty models for strategies and beliefs about those strategies • Inception levels are recursive (pushdown) automata module calls • Generalized risk profiles (in virtuality) for strategies are considered Inception Games
  7. 7. 1. Coalition teams/agents (inceptors and inceptees) [reverse or anti-inceptions are possible] 2. Inception information silos 3. Payer source (client) and payoff structure for agents (to be distributed ) 4. Inception dream levels and physical/psychical rules of engagement (time-discounted recursions) 5. Consciousness awareness threshold for inception advantage (to be discussed) 6. Uncertainty structure (stochastics) for consciousness awareness and social power status for agents 7. Risk profiles (spectral range) of agents and collective coalitions and risk nbhd. size thresholds 8. Coalition bonding thresholds (propensity of agents to n-agency behavior) Components of generalized inception game inception information silo agent’s portion of inception information agent i agent j coalition kC coalition lC inception demarcationinception coalitions inceptee coalitions anti-inception information silo $ payer source ………………. Inception levels Inception Games
  8. 8. Decision Branching and Game History Trees Inception Games . . . . . . 1a 2a 2a 3a 3a 1 1d 1 1 k d . . . . . . 1 2d 2 2 k d ………. ………. ………. 1na  na na 1 nd nk nd hypermatrix of agent payoffs at stage j i j jp s . . . . . . . . . . . . . . . decision branches extensive form game branching translates to payoffs and agent information sets information set for agent k,  r h k   :k rH h k r  empty cells in hypermatrix mandate next stage choice Show payoffs for all possible strategies . . . . . . . . . . . . . Belief revision operators are applied to history sets l rB h k  
  9. 9. Description of Inceptions as Strategic Recursive Games Strategic game in which successive levels of “insider information” can be obtained in compacted real time. Akin to computationally fast what-if risk assessments and decision analysis using projected data (multiple-universe scenarios), run by generalized simulations. (hybrid human/machine games). More importantly, the question, “is knowledge of being in an inception attempt a strategic (psychological) advantage in an inception level if other agent inceptees do not know they are in one?”, is discussed as a game strategy dynamic. Game-theoretic analogies through ad infinitum strategy second guessing or strategy regress and recursive time-discounted stochasticity in inceptions. An inception may represent recursive belief revisions. Emergent computational models may simulate dynamics of recursive games of this nature (universal belief conflicts). Inception Games   1,...i i 
  10. 10. Recursion in Inceptions Inception Games In Inceptions (recursive games), terminal payoffs occur in an entrapped inception level (limbo death) or are bumped up to reality level (inception). Each level that is ascended is assumed to have a stable agent that can handle that ascension (PASIV machine operator) and so, a chain of coalition agents must be in place to ascend to reality level. This introduces a new dynamic for a game with dominantly protected agents (ambassadors) at each recursive inception level. Binary inceptions Partial, incomplete, imperfect inceptions Inception with GTU-based distribution, 1g 2g lg
  11. 11. Some Research Questions • Can inception-like rules in a game-theoretic setting emulate or improve real world decision making and strategies in organisms? • Are there intrinsic novelties in the game structure of inceptions or do they fall into a category of strategy models or patterns in game theory with certain useful and practical solutions/equilibria ? • Can one frame belief revision (operators) on belief systems of coalitions for inception? • Using a virtual world(VW)/holodeck, can hybrid human-automaton multi-agent systems form (emergently) superior strategies not predictable using classical rational decision/game theory? • Can risk be a generalized multi-dimensional measure beyond intervals or singletons and can it be translated to the human as sensory stimuli? • Are (inception) game structures representative of a more higher-order abstract such as a logic, automata, or categories/topoi? Inception Games
  12. 12. Theories, Concepts, and Hypotheses • Inceptions are equivalent to general social stochastic (dynamic) recursive games with individual/coalition strategies, behavioral rules, belief revision, utility functions, and compacted time-discounted evolution. • Mathematical modeling and simulation of inception-like behavior in virtual gaming is a means to emergent and evolutionary strategic gaming scenarios producing advanced predictive (risk) analytics [built into equivalent advanced game analytics]. • Inception-like (dream) levels may be emulated in modeled virtual worlds to simulate emergence of strategic behavior in generalized conflicts involving coercive information flow. • Through inception dynamics and novel representations, generalized decision risk can be effectively and directly linked to human-machine sensorium [enhanced hair on the back of your neck]. • Inception games (dynamic games) have more powerful abstractions and higher-level mathematical representations as categories/topoi/logics/automata. Inception Games
  13. 13. Limitations • Computational complexity of search and/or calculation of solutions • Massive numbers of agents (and coalitions) • Assumptions on uniformity and consistency of thresholds and distributions of beliefs • Recursions may be self-referential without mechanism for checking • Simulations of human play are gross approximations to spectrum of human bounded rationality • Strategy regress and n-agencies may introduce instabilities and chaos into regions of belief • No human players • May not like answers: no stable solutions to inception conflicts – need to make semi-unrealistic assumptions • Mixing belief uncertainties among agents may lead to a type of belief neurosis or regress • Mapping generalized risk to multiple human senses is limited by current technologies, ethics, and low thresholds of cognitive (sensory) overload – few results on holodeck-itis effects. • Theory-laden study with no prior literature results on inceptions or generalized conflicts Inception Games
  14. 14. Research Methodologies Grounded theoretical framing (theory and concept building) based on (i) emergent game theories , (ii) generalized uncertainty metamodels, (iii) the analogies of inceptions as recursive decision branches, (iv) higher-level mathematical abstractions of games, and (v) virtualization of risk in decisions and games. May consider design of science methodology of simulation-based emergence of models of belief revision within strategic (inception) games ala Markov Chain Monte-Carlo (MCMC) simulations of model building of strategy profile distributions and beliefs about them. Consider information criteria (IC) approach to finding optimal (parsimonious and high generalizability) model sizes and/or model families of assessments for belief revisions. For families of Bayesian belief revision models , Inception Games             * * , argmax , , , , d d M d N d d M IC M N IC M N l M d N                dM  function of sample size defining type of IC statistic dimension of parameterization likelihood of modelIC statistic penalty term parameterization of model model
  15. 15. Data Collection and Analysis Simulated game playing results (game analytics) from inception modeling in VWs/holodecks [proposed for follow up studies]. Apply dynamic strategy beliefs as general uncertainty distributions on strategy profiles, starting from uninformative uniform priors, evolving to more informative updated priors as game proceeds. Collect stage terminations, payoffs, histories, and belief revisions used. Glean any meta-patterns of strategies and payoffs based on high probability repeats from inception thresholds. Inception Games
  16. 16. 1 1 j m n C i j i R R R     Agent [coalition] payoffs , are given in terms of the relative worth of: (i) the agent’s role in the extraction of, and (ii) the portion of, the total inception information , at time t, utilizing action space : where,    ji C iU A U A    iA             , Z , , ji i j j i i i i ji i i j Ca a i C t i i i t i i C a A a A Ca a ak i i t t t t i t t t k C U A Z g a U A Z g a Z Z Z f R Z f R            tZ I : , :i if R I f R I  extraction functions acting on agent resources, producing information subsets Coalition cumulative resources are assumed to be comprised of total agent resources (information silos): ia i tZ extracted inception info as a result of action , at time t (stage) for agent iia ji Ca tZ extracted inception info as a result of actions , at time t (stage) for coalition jia Inception Games Inception Payoffs  ig a GTU-based uncertainty operator on action ia
  17. 17. Strategies are mixed action profiles through stages of game transitions: Agent actions , are applications of belief revision operators to the belief systems of the opposition in inception:   1 ... :k j j jjj u i i i i C Ca B B     updated belief system belief system of coalition jC composition of belief revision operators  1,...,b ,...i ks b k-th stage profile action is a GTU-based mixture     1 kL k i j j j j b g A w g a     generalized uncertainty function (GTU-based) of agent action space i a weighted sums of GTU distn. actions in simple linear case Inception Games Belief Revision of Agent Strategies
  18. 18. Inception Games Belief systems in games can be viewed as agent beliefs about past histories of moves. These past decision histories, , are a component of finite extensive games, along with an equivalence relation for those histories that defines history classes where actions are taken without regards to a history in that class. An assessment is a pair , where is a strategy profile and is a belief system for a game. Belief systems are updated (using a Bayesian procedure or GTU-based updating scheme) by the information sets ,of the agent’s accessible nodes in the equivalent game decision tree. Each information set , denotes the sum total of the agent’s knowledge of the universe. Compatibility between and can be quantified by the concept of KW-consistency – there exists an infinite sequence of mixed strategy profiles such that where is the updated belief system of which is associated with the strategy profile . iH  i i N   ,    iI iI     1,...i i      lim , ,u i i i       u i i i Belief Revision of Agent Strategies
  19. 19. Inception Games A more recent and practical type of compatibility is given by AGM-consistency which is associated with the idea of the AGM belief update revision framework. Based on plausibility order in the space of all agent histories . Plausibility orders are total pre-orders on sets, (i.e., binary relations which are complete and transitive ). is AGM-consistent if : (i) and (ii)   1,...,i i N H H   H H   ,    0 ~ [w.r.t ]a h ha      , 0 ', 'h D h h h h I h      history seq. h followed by action a set of histories with same information set as h. prob. of action a assigned by prob. of history h assigned by  Belief Revision of Agent Strategies
  20. 20. Inception Games If makes the assessment , AGM-consistent, then is said to rationalize . is said to be Bayesian relative to if for every -equivalence class E, there exists a prob. meas. such that (i) , and (ii) and (iii) for every information set I such that and for every , where . An assessment is Bayesian AGM-consistent if it is rationalized by a plausability order on H and it is Bayesian relative to . is a perfect Bayesian equilibrium if it is Bayesian AGM-consistent and sequentially rational. perfect Bayesian equilibrium subgame-perfect equilibrium sequential equilibrium perfect Bayesian equilibrium Hence perfect Bayesian equilibrium is intermediate between subgame-perfect and sequential equilibria. Every finite extensive-form game G, (and hence every finite decision game tree ) has at least one perfect Bayesian equilibrium .  ,   ,   ,   : 0,1E H   ESupp E  if , ' and a prefix of ',thenh h E h h        ' ... miE Eh h a a       Min I E         , | E E E h h I h h I I         : ', 'h I h h h IMin I     ,   ,     G  ,  Belief Revision of Agent Strategies
  21. 21. Belief revision operators , act on propositions (sentence contents) in a belief (logic) system B, by one of three broad categories: 1. Expansion 2. Revision 3. Contraction Revision and contraction require belief revision operators to be minimally invasive, (i.e., conserve as much as possible, logical consistency within B according to epistemic AGM-consistency framework. Let denote set of formulas in a propositional language L which is based on a countable set of atoms, S. Subsets have deductive closures denoted by . Closed if and consistent if . Formally, an agent’s initial belief system is consistent and closed but is exposed to subsequent information given by . A belief revision function based on K is a function such that . If If satisfies AGM postulates*, then it is an AGM belief revision function. u B  B   Inception Games  K    K  K K  K      : 2KB      ,KB     updated revised belief system new information formula ,partial belief revision , full belief revision       KB * see Bonanno (2011) for postulate details Belief Revision of Agent Strategies
  22. 22. Inception Games However, one must reconcile belief revision syntactically in propositional languages with that in set-theoretic game structures. Use choice frames (from rational choice theory) to link the two and then form belief revision operators. Choice frame triplets : interpret as available alternatives (potential information) and as the chosen alternatives which are (doxastically or believably) possible. Use Condition C: , where D is profile decision space and is profile decision space for agent i (no consecutive actions by any agent). In case of inceptions, one can composite a series of actions into one actionable move. Associate models with choice frames through valuations of atomic formulas , maps formulas to the states that are true under them. Model is quadruple and is an interpretation of the choice frame .  , ,C f      , set of states, subsets are events, 2 , collection of non-null events ; 2 , function associates events with non-null events satisfyingf f E f E E          , , , if theni ii N h D a A h h D ha D        iD E   f E : 2v S    , , ,M f v   , ,C f  Belief Revision of Agent Strategies
  23. 23. Inception Games Choice frame is AGM-consistent if for every model M based on it, partial belief functions associated with M, can be extended to full belief functions that are AGM belief functions. Here , . , and means is true at state in model M. Choice frame C, is rationalizable if there exists a total pre-order on such that for every , . Build partial belief revision funcs: is AGM-consistent C is rationalizable. Let denote the set of total pre-orders that rationalize a game choice frame and additionally satisfy conditions PL1 and PL2i of Bonnano (2011). Define a game common prior by (common initial beliefs and disposition to change those beliefs in a game setting) A game choice profile given by admits a game common prior if there exists a total pre-order on H that rationalizes the beliefs of all agents and satisfies the conditions PL1 and PL2i for each agent, (i.e., ).  , ,C f  MK B  E     : ', 'f E E E       at least as plausible as most plausible states in E  , ,C f     , ,i i i N H f  agent game histories i   , ,i i i N H f  i i N    Beliefs and dispositions to change those beliefs about game strategies   :M M K f        truth set of formulas : | MM M         | M          : 2 , : , :M MK M M KM M M B B f               Belief Revision of Agent Strategies
  24. 24. Inception Games Let an extensive form game (and hence a game tree) satisfying condition C by given by G . Let be a profile of AGM-consistent choice frames for the initial beliefs and beliefs on changing them for all agents. If admits a common prior (i.e., ) then every common prior , is a plausability order and hence, a belief revision operation. More generally, belief revision operators , based on GTU-based constraints g, acting on beliefs B, represent very general belief revision uncertainty schemes such as Dempster-Shafer, Zadeh possibility distns., quantum probs., quantum-gravity causaloids, Bayesian causal nets, fuzzy beliefs, rough set beliefs, classical probs., first-order and higher order logics, including paraconsistent systems, etc. Composited GTU-based belief operators can be constructed to form generalized belief revision operators where each operator satisfies conditions of AGM-consistency and common prior above:   , ,i i i N H f    G   , ,i i i N H f    p g         1 1 1 ... : ,..., k j j jj u i i i k C C k a g g g B B g g g     GTU constraint vector scheme for k cascaded (composite) agent actions Belief Revision of Agent Strategies
  25. 25. Belief systems B, which are knowledge bases of propositions are updated more precisely, based on GTU-based belief revision operators , using generalized likelihood-type transformations , that utilize updated information at stage k, , as in Bayesian approaches to posteriori probability distributions. A proposition , which can be treated as a general uncertainty distribution or rule itself, is then transformed to another rule/distribution by .    i ig g    g g g B  kI Chained compositions of belief revision operators can be interpreted as recursive likelihoods, (i.e., given updated information in each inception level, a chain of uncertainties about uncertainties are generated. We refer to these chains as higher-order beliefs. Inception strategies are then profiles of GTU-mixed actions which are chained compositions of GTU-based likelihood-type transformations of belief system propositions. Given the threshold , defining a safety margin from consciousness awareness advantage of one agent over another or of a coalition agency, computing inception game equilibria (or any other equilibria) resembles a Schilling type of segregation model dynamic. Recent results show that segregation or end game convergence depends on neighborhood sizes and psychological segregation thresholds. Inception neighborhoods are those areas that influence an agent directly from individual agent and coalition inceptions and information exchanges from such.  Inception Games Belief Revision of Agent Strategies
  26. 26. Agent belief revision operators are dependent on agent (and coalition) social power status which is in turn, indicated by consciousness-awareness, the relative ability to know that you are in a certain conscious (inception) level, while others are not. Probabilistically (generalized to GTU constraints ): Agent i revises the belief system of agent k in inception level j if for some threshold . induces preference ordering for belief revision operations and hence for strategy execution. Based on social power preference, a weighing of influence will affect extraction of the portion of the inception information from an agent’s influence neighborhood through belief revision operators acting on inceptee’s belief system and hence on the consciousness-awareness uncertainties .  ,iG j k  prob. (GTU uncertainty) agent (avatar persona) i knows consciousness awareness of agent k in agent i’s inception level j    , ,i kG j k G j i        , , , ,iG j k i j k Inception-like Games Belief Revision of Agent Strategies i k inception level j iL  ,iG j k            1 ? 1 ... : , , ,vjj g i i i v i i ka g g g G j k G j k G j i      iG
  27. 27. The knowledge that an agent is aware of an inception attempt happens when prob. of consciousness awareness over another agent is within threshold (but not greater) and other agent attempts inception against that agent: Akin to asymmetrical agent information, biases against any attempted belief revision, no matter the truth value, T (or truth membership value) of the belief revision operator(s) update on B of inceptee. Hence, the apriori agent knowledge of an outside inception weighs against any belief revision attempt, regardless of belief revision update consistency. On the other hand,, if an inception is anticipated, but no such attempt is made, agent is less likely to accept useful information from other agents in information exchanges pertaining to other inceptions (intra-coalition exchanges).      , , ,k i kG j i G j k G j i     Inception-like Games Consciousness-awareness and social order in coalitions: The Power to Change Minds
  28. 28. Inception-like Games Epistemic effects from inceptions include trustfulness vigilance - the propensity to initially trust and follow through with stronger verification in the later stages of interaction and decision-making regardless of trusting history successes. Gilbert (1991) posits that a Spinozan unity doctrine , of initial belief in a proposition is necessary in order to commence understanding (and veracity) of that proposition in a belief system. Moreover, doubt or unacceptance is harder to materialize than acceptance. More recent research has shown that a suspended belief - conditional Cartesian ? - is more likely in an initial understanding of propositions with informative priors (Hasson, Simmons, and Todorov, 2005). A hybrid Cartozan framework in which comprehension followed by temporary acceptance and then possible unacceptance is also posited. Epistemic vigilance may be more likely following brief initial acceptance after comprehension. We propose that an evolutionary process takes place that manifests in possibly chaotic or recursively fractal regions of attraction between and as belief revision meta-theories for GTU-based belief revision operators . We pick GTU operators because of their generality in representing vast diversities of logic systems for uncertainty. g  Validating Beliefs and Pre-comprehension: Spinoza to Cartesian
  29. 29. Inception-like Games Understanding a proposition is more mechanical (e.g., computing complexity measures) than believing it (e.g., belief proposition is (AGM-)consistent within a belief system). We consider the inception model as a meta-type for a game involving recursive belief revision through time-discounted inception levels and application of GTU-based belief revision operators to the belief systems of the coalition agents involved in an inception attempt. Coalitional belief equilibria are possible under conditions of informative common priors on agents initial belief systems and dispositions to change those beliefs through assessments which are AGM-consistent within the extensive form of the inception game. -inceptions are more likely, perturbing equilibria between regions of and degreed priors. - priors lend to belief revision operators that initially support, with certitude, the truth value of the status quo propositions being challenged in inceptions. priors lend to initial support , with certitude, the false value of status quo propositions in inceptions.  ,    Validating Beliefs and Pre-comprehension: Spinoza to Cartesian
  30. 30. In a GTU-based belief revision regime, a spectrum of initial belief in status quo propositions in inceptions is created based on the common prior distns.’ informativeness of those propositions. We denote this spectral measure of epistemic belief doctrines by . This measure may be a complex of objects rather than a simple scaled number. We can then create an approximate simplifying normalized number based on this complex , that measures a divergence of the ensuing initial prior support for status quo propositions that govern agent beliefs about other agents’ strategies from an or -like doctrine. In infinite games such as conceptual inceptions, strategy near-solutions may cycle in regions in the spectrum. The inception information sets of the agents may dictate certain regions of attraction and hence of certain belief doctrine spectral subregions. Common informative priors of agents are candidates for inceptions and for manifesting well-defined belief doctrine spectral measurements.   S C  S C Inception-like Games Spinozan Cartesian spectrumS C  S C  Pre-comprehension belief Validating Beliefs and Pre-comprehension: Spinoza to Cartesian
  31. 31. How does one measure belief revision potential on the spectrum? In inceptions, the end game value (payoff) is the information that is sought after and an ensuing changing of the “hearts and minds” of the inceptee. Assume the initial belief systems of the coalition agents are the target. Belief revision operators are applied to those belief systems. The common prior to those initial beliefs is then the baseline for belief revision movement. A belief revision operator or subsequent prior to an agent’s belief system then results in an updated belief system. In a Spinozan doctrine, those belief revisions are not changed as much (if at all) as those from a Cartesian doctrine because acceptance of the initial beliefs is more likely from a Spinozan than a Cartesian or even a Cartozan. Therefore, a metric that would measure a distinction of this initial movement would be a divergence between priors after and before a belief revision operation. A candidate for this divergence would be a Csiszar f-divergence which is a generalization of the KL-divergence between probability measures. We generalize this divergence to measure differences between GTU-based uncertainty operators: S C Inception-like Games Measuring Epistemic Belief Revision Potential
  32. 32. Inception-like Games Define a Csiszar-Morimoto-Ali-Silvey (CMAS)-generalized divergence between a initial prior and its belief revision update for each : and u  , ,T f w u u g g u p dp D tr wf dp dp             , ,f w u u g g u dp D p p tr wf dp dp           p u p An accumulated (total) divergence between prior spaces , may then be formed as: p This total divergence may then be symmetrized to form a metric using a weighted Jenson-Shannon divergence scheme:        , , , , , 1 ,T f w u T f w u T f w u g g gD D D      for 0,1 .  Measuring Epistemic Belief Revision Potential
  33. 33. Inception-like Games Define a unit norm on the space of possible belief priors : where is the belief system updated after uniform pdfs are applied as belief revision operators to all individual agent priors in . One may then define a normed spectral measure for on by applying the unit norm above: so that for any .        , , , sup , T f w U g T f w g D D      U  S C     , 0 1    Measuring Epistemic Belief Revision Potential
  34. 34. Inception-like Games Updating mechanism is most popularly manipulated using Bayesian rules. Is Bayesian updating optimal in any sense besides probabilistic learning? Consider the use of reward optimization models using infinite partially observable Markov Decision Processes (iPOMDPs) as alternative where actions are of two types, (i) reward seeking and (ii) knowledge seeking. General iPOMDPs to GTU-based uncertainty operators where mixed uncertainty regimes are utilized for uncertainty among agent action choice, payoffs, state transition, and belief updating about those uncertainties. . Uncertainty Updating of Priors: Generalizing Bayesian Learning
  35. 35. Inception-like Games Correcting for better calibration of probability-calculation by agents for priors may be done by: 1. under-scoring precision of agents from past histories and calculated updating revisions 2. present certain amount of additional alternative beliefs in the space of priors to agent information sets (normalize confidence) (Mannes and Moore, 2013). 3. Inception priors may be more precise by flattening agent prediction egos. 4. Use diversity of uncertainty operators for a larger amount of situational epochs, (i.e. use the full power of GTU constraint representations, mixing, and application) Need to form a real-time or stage-updated comprehensive and multi-dimensional risk measurement object during game play that takes into account aspects of uncertainty operators and scenarios, complete information sets, and consistent calibration of agents as decision-makers (DMs). This risk object must be communicated to DMs in a more natural, ubiquitous, and instantaneously comprehensive manner. Next, we discuss a multi-sensorial approach to this risk representation to DMs. Making For Better Decision Makers and Judges of Truth Values
  36. 36. Risk measurement and visualization of inception strategy dynamics Overall risk metric (risk manifold ) is multi-dimensional assessment of risk-related components, including: 1. Expected utility values at stages or time epochs (w.r.t. GTU-based uncertainty operators) 2. Psychologically (or otherwise) assigned belief weights to utilities, uncertainties, and payoffs 3. Coalition partition preferences (who do you want to prosper or lose regardless of your situation) 4. Risk tolerance - thresholds on a spectrum of risk-aversion/aggressiveness [fear/confidence] 5. Statistical and computational error tolerances of quantitative risk measurements 6. Targeted Lipschitz game effects on agents or groups (similar to 3. above) 7. Size and makeup of coalitions 8. Risk capacity – relative resource reservoir (loss absorption capacity) 9. Time horizon of play endurance (time expanded fluctuation and volatility smearing) 10. Risk efficiency (vs. utility), also called efficiency frontier (MPT curve of optimal utility vs. risk) 11. Confounded effects between the above factors (i.e., resource size vs. risk tolerance) Inception-like Games Expanding and Expressing Risk in Games and Decisions Making
  37. 37. Novel approach to uncertainty in risk: Generalized risk measurement through composition of belief revision operators (likelihoods) that result in cascaded GTU-based operator (diversity and recursion of uncertainty types): based on k-th stage information extract (partial inception) , updated belief system , and composite belief revision operator (action taken) .               11 1,..., ,..., ... : , E L , | k k k u gg g g g g u u g g g kg g R B B I                  kI u gB  g (generalized risk operator) Inception-like Games Expanding and Expressing Risk in Games and Decisions Making
  38. 38. Map isomorphism between components of risk manifold and human sensorium manifold Graphics using diverse visualization only one component of human measurement Need to utilize a functional subset of human sensorium (i.e., olfactory, haptics, proprioception, thermal) Consider DIY holodeck-type approach in depicting this functional subset of DIY Holodeck setup Conceptual risk-sensorium map Inception-like Games Naturally Sensing Risk: Virtual Hair on the Back of Your Neck
  39. 39. Need richer visual representations of risk analytics and quantitatives, consider topological/geometric props instead of static graphical tools such as traditional 1D/2D linear colormetrics Proposed a visual generalized 4-D prop class named i-morphs that is dynamic and responds to touch, speech, and other human sensorium interaction channels. Dimensional sizes, shape, dynamic behavior, color, and other attributes of i-morphs translate to quantitative risk components of an object in risk manifold , of inception game stage or history of stages. Inception-like Games What Does Risk Look Like?
  40. 40. Use recursive (pushdown) automata theory to compute game equilibria using recursive module calls. Game solutions are inception or near inception states, -inceptions which are -Nash equilibria or in case of evolutionary games, -evolutionarily stable solutions. These solutions may also be referred to as belief equilibria since the end game is coercion of belief revision manifested in an inception (extraction of information).    Inception-like Games Inceptions are Recursive Automata: Computation of Game Solutions
  41. 41. Coalition belief revision patterns may enter chaotic regions (attractors) between subregions of near inceptions and near anti-inceptions (reversal of inceptions). Coalition agents may evolutionarily become n-agencies (agent siding regress between coalitions). Thresholds , determine belief equilibria dynamics, similar to segregation models of Schilling, in which nbhd. sizes (size of a nbhd. of like-minded agents) , are also relevant to equilibria.   Inception-like Games Belief Regions, Solutions and Equilibria
  42. 42. Features of Inceptions • Recursive stochasticity (generalized to GTU constraints) • Time deformation (compacted) levels (equivalent to time-discounted subgames/subautomata) • General recursive pushdown computational machines/automaton • Graphical tree representations through normal form game equivalents • Generalized social conflict games • Evolvable (evolutionary pushdown automata and games) • Can model risk dimensions within game dynamics • Representable by general uncertainty models, logics, and higher level abstractions • Can use physical models (quantum, quantum-gravity, algebraic categories) for dynamics • Model belief revision dynamics and equilibria Inception-like Games
  43. 43. Discuss, review, and postulate alternate model components applicable to inception game structure in: Appendix A: Classical and non-classical decision and game theories Appendix B: Emergent game theories, including: Differential, stochastic, evolutionary, continuum, and stopping games Generalized social games (complexes of rules) Quantum-like games Large-scale hypergames, abstract board games, and computational issues Morphogenetic approaches to computational games, including Shilling models Abstract economic games Hybrid games Lipschitz games Inception game categories and topoi (see Appendix E for preliminary details) Boundedly rational and emergent forms of utility theories Appendix C: Quantum-gravity causaloid (generalized physical causal net) approach to uncertainty Appendix D: Zadeh’s General Theory of Uncertainty Appendix E: Category and Topos Theory and Game Representations Inception-like Games Supporting Content for Inception Game Concepts
  44. 44. Future Considerations • Simulate inception game with risk components based on a large-scale game representation in a DIY physical holodeck or virtual world simulation such as SL. • In simulations, substitute real-world scenarios using mixture of classical and non-classical uncertainty operators via GTU-based constraint representation in inception game. • Generate diverse belief revision operators and thresholds for local inceptions (agent-to-agent). • Develop graphical tree-like representations for belief revision-based dynamics for normal-form of inception game. • Simulate a quantum-gravity interaction dynamic based on causaloid nets in inception games. • Develop more efficient computational models for searching/calculating game solutions for inceptions (i.e., Nash-like and evolutionarily stable equilibria). • Develop graphical display models to simultaneously visualize/sense local (micro or super- micro) interactions with (super) macro and meso-level dynamics in inception games. Inception-like Games
  45. 45. Conclusions • Inception games are a conceptual abstraction for generalized physical-social interaction with resource exchange (inception and other utility tradeoffs). • Diversity of uncertainty models and evolutionary dynamics can be included in inception game description and representations. • Simulations may be run based on mapping game dynamics with real-time risk sensorium for the decision-maker or decision-making coalitions involved in the risk theatre. • Multi-dimensional and sensorial holodecks are ideal tools for mapping complex game dynamics to decision-making entities via risk connectives. • Inception game generalities are representable by higher order meta-mathematics such as category and topos theory, automata logic theory, and biologics through risk sensorium mapping. • Inceptions are novel ways to interpret general complex conflict scenarios independent of domain of application (i.e., military maneuvering, business ecocycles, government-social interaction, etc). • Lack of detailed simulation algorithm results in study to be accommodated in simulation /algorithm development, testing, collection of results, and comparative analysis in follow up studies. Inception-like Games