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.

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. 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. 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. 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. 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. 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. 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. Decision Branching and Game History Trees
Inception Games
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1a
2a
2a
3a
3a
1
1d
1
1
k
d
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1
2d
2
2
k
d
……….
……….
……….
1na 
na
na
1
nd
nk
nd
hypermatrix of agent payoffs
at stage j i
j jp s
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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
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Belief revision operators are applied to history sets l rB h k  

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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