# Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Comparison with Related Work

#### 1.2. Applications of Multigames

#### Social Interactions and Sub-Personalities of Agents

## 2. Bayesian Games and Their Ex Post NEs

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Definition**

**2.**

**Definition**

**3.**

**Proposition**

**2.**

#### Outline of the Paper

## 3. Linear Multidimensional Bayesian Games and Uniform Multigames

**Definition**

**4.**

**Definition**

**5.**

- 1.
- The set of agents is $I=\{1,\dots ,n\}$.
- 2.
- The set of n-agent basic games is ${G}_{j}$, where $j\in J=\{1,\dots ,m\}$ with action space ${A}_{ij}$ and utility function ${U}_{ij}$ for each agent $i\in I$ in the game ${G}_{j}$.
- 3.
- Agent i’s strategy is ${s}_{i}=({s}_{i1},\dots ,{s}_{im})\in {S}_{i}={\prod}_{j\in J}{A}_{ij}$ where ${s}_{ij}$ is agent i’s action in ${G}_{j}$.
- 4.
- Agent i’ type is ${\theta}_{i}=({\theta}_{i1},\dots ,{\theta}_{im})\in {\mathsf{\Theta}}_{i}$ with ${\theta}_{ij}\ge 0$, ${w}_{i}>0$ and ${\sum}_{j\in J}{\theta}_{ij}\le {w}_{i}$.
- 5.
- Agent i’s utility for the strategy profile $({s}_{i},{s}_{-i})$ and type profile $({\theta}_{i},{\theta}_{-i})$ depends linearly on its types:$${U}_{i}({s}_{i},{s}_{-i},{\theta}_{i},{\theta}_{-i})=\sum _{j\in J}{\theta}_{ij}{U}_{ij}({s}_{1j},\dots ,{s}_{nj}).$$
- 6.
- The agents’ type profile $({\theta}_{1},\dots ,{\theta}_{n})\in {\prod}_{i\in I}{\mathsf{\Theta}}_{i}$ is drawn from a given joint probability distribution $p({\theta}_{1},\dots ,{\theta}_{n})$. For any ${\theta}_{i}\in {\mathsf{\Theta}}_{i}$, the function $p(\xb7|{\theta}_{i})$ specifies a conditional probability distribution over ${\mathsf{\Theta}}_{-i}$ representing what agent i believes about the types of the other agents if its own type were ${\theta}_{i}$.

**Proposition**

**3.**

- 1.
- The set of n-agent basic games is ${\widehat{G}}_{j}$ where $\widehat{J}=J\cup \{m+1\}$ with: (i) action space ${A}_{ij}$ and utility function ${\widehat{U}}_{ij}$ for each agent $i\in I$ in the game ${\widehat{G}}_{j}$ with ${\widehat{U}}_{ij}={w}_{i}{U}_{ij}$ for each $j\in J$; and (ii) action set ${A}_{i(m+1)}={A}_{i1}$ and utility function ${\widehat{U}}_{i(m+1)}({s}_{i(m+1)},{s}_{-i(m+1)})=0$ for each ${s}_{i(m+1)}\in {A}_{i(m+1)}$ and ${s}_{-i(m+1)}\in {A}_{-i(m+1)}$.
- 2.
- Agent i’s type space,$${\widehat{\mathsf{\Theta}}}_{i}=\left\{\left(\frac{{\theta}_{i1}}{{w}_{i}},\dots ,\frac{{\theta}_{im}}{{w}_{i}},1-\frac{{\sum}_{j\in J}{\theta}_{im}}{{w}_{i}}\right)|({\theta}_{i1},\dots ,{\theta}_{im})\in {\mathsf{\Theta}}_{i}\right\}.$$
- 3.
- Agent i’s total resource is ${\widehat{w}}_{i}=1$.

**Proof.**

**Proposition**

**4.**

**Proof.**

#### Equivalence of Linear Multidimensional Games and Uniform Multigames

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

## 4. Standard Uniform Multigames and Their Compatible Pure Ex Post NE

**Definition**

**7.**

- 1.
- Agent i’s action set ${A}_{i}$ consists of an action denoted by ${s}_{ij}$ for each $j\in J$. The actions ${s}_{ij}$ for each $j\in J$ are not necessarily distinct and thus ${A}_{i}$ contains at most m actions.
- 2.
- The strategy profile $({s}_{1j},\dots ,{s}_{nj})$ is a NE for the basic game ${G}_{j}$ for each $j\in J$.

**Definition**

**8.**

**Definition**

**9.**

#### 4.1. Running Example: A Double Game for Prisoner’s Dilemma (PD)

#### Numerical Examples of Computation of Ex Post NE

**Example**

**1.**

**Example**

**2.**

## 5. Pure Ex Post NE Induced From Type Profile Space Boundary

#### 5.1. Partition of Type Profile Spaces

**Proposition**

**5.**

- (i)
- ${\mathsf{\Theta}}_{i}={\bigcup}_{1\le j\le m}{\mathsf{\Theta}}_{ij}$ with ${\mathsf{\Theta}}_{ij}\cap {\mathsf{\Theta}}_{ik}=\varnothing $ for ${s}_{ij}\ne {s}_{ik}$ and ${\mathsf{\Theta}}_{ij}={\mathsf{\Theta}}_{ik}=\varnothing $ for ${s}_{ij}={s}_{ik}$ for $j,k\in J$;
- (ii)
- ${p}_{j}\in {\mathsf{\Theta}}_{ij}$ for $1\le j\le m$; and
- (iii)
- for all $\theta \in \mathsf{\Theta}$, the local game ${G}_{({\theta}_{1},\dots ,{\theta}_{n})}$ has as a NE the action profile $({s}_{1{j}_{1}},\dots ,{s}_{i{j}_{i}},\dots ,{s}_{n{j}_{n}})$, where ${j}_{i}$ is given by ${\theta}_{i}\in {\mathsf{\Theta}}_{i{j}_{i}}$ for $1\le i\le n$.

**Proof.**

#### 5.2. Partition of an Agent’s Type Space Given Other Agents’ Types

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

- 1.
- ${\mathsf{\Theta}}_{i}={\bigcup}_{j\in J}{\mathsf{\Theta}}_{ij}\left({\theta}_{-i}\right)$ where ${\mathsf{\Theta}}_{ij}\left({\theta}_{-i}\right)\cap {\mathsf{\Theta}}_{ik}\left({\theta}_{-i}\right)=\varnothing $ for $j,k\in J$ with $j\ne k$; and
- 2.
- given ${\theta}_{i}\in {\mathsf{\Theta}}_{i}$, the local game ${G}_{({\theta}_{i},{\theta}_{-i})}$ has a NE for a strategy profile $\left({s}_{i}({p}_{j}),{s}_{-i}({\theta}_{-i})\right)$, where $j\in J$ is the game for which ${\theta}_{i}\in {\mathsf{\Theta}}_{ij}({\theta}_{-i})$.

**Proof.**

**Definition**

**10.**

**Example**

**3.**

**Example**

**4.**

#### Main Results and Algorithm

**Theorem**

**2.**

- (i)
- $({s}_{1}(\xb7),\dots ,{s}_{n}(\xb7))$ is a compatible pure ex post NE for G on ${\mathsf{\Theta}}^{b}$.
- (ii)
- For each agent $i\in I$ and ${\theta}_{-i}\in {\mathsf{\Theta}}_{-i}$ there exists a partition ${\left\{{\mathsf{\Theta}}_{ij}\left({\theta}_{-i}\right)\right\}}_{j\in J}$ of ${\mathsf{\Theta}}_{i}$ with ${\mathsf{\Theta}}_{i}={\bigcup}_{j\in J}{\mathsf{\Theta}}_{ij}\left({\theta}_{-i}\right)$ such that the set ${\mathsf{\Theta}}_{ij}\left({\theta}_{-i}\right)$ is independent of ${\theta}_{-i}\in {\mathsf{\Theta}}_{-i}^{e}$ for each $j\in J$.

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

Algorithm 1: The test for the existence of a pure ex post NE |

## 6. Pure Ex Post NEs in Multimarkets

#### 6.1. Adoption of Technology

#### 6.2. Multimarket Production

## 7. Multi-Games with Multi-Stage Basic Games

**Example**

**5.**

**The Trust Game**Assume a two-agent stage game ${G}_{1}$ in which ${A}_{1}=[0,1]$, ${A}_{2}=\left\{x\right|3y\ge x,y\in {A}_{1}\}$ and ${u}_{1}(y,x)=x-y$, ${u}_{2}(y,x)=3y-x$ for $y\in {A}_{1}$ and $x\in {A}_{2}$. By backward induction, when the first agent plays first, $(0,0)$ is the Nash equilibrium (NE). If, for the sake of illustration, we restrict Agent 1’s actions to ${A}_{i}^{\prime}=\{0,1\}$ for $i=1,2$, then Figure 5a shows the branches of the stage game where the two agents are named ${a}_{1}$ and ${a}_{2}$, respectively. As usual, the label on each edge is the action taken by the agent on the node above and, under each leaf, the first number is the utility of Agent 1 for the branch corresponding to the leaf and the second number is Agent 2’s utility.

**Example**

**6.**

**Double game for Trust Game**Let ${G}_{1}$ be the Trust Game as in Example 5 and let ${G}_{2}$ be the associated conscience game in which ${A}_{1}=[0,1]$, ${A}_{2}=\left\{x\right|3y\ge x,y\in {A}_{1}\}$ and ${u}_{1}(y,x)=y$ and ${u}_{2}=x-2y$ for $y\in {A}_{1}$ and $x\in {A}_{2}$. By backward induction, $(1,3)$ is the NE. If again, we restrict Agent 1’s actions to ${A}_{i}^{\prime}=\{0,1\}$ for $i=1,2$, then Figure 5b shows the branches of the stage game. Consider a double game G with basic games ${G}_{1}$ and ${G}_{2}$ where ${A}_{1}=[0,1]$, ${A}_{2}=\left\{x\right|3y\ge x,y\in {A}_{1}\}$, ${\mathsf{\Theta}}_{1}=\{1/4\}$ and ${\mathsf{\Theta}}_{2}=\{0,2/3\}$. We have ${U}_{1}(y,x,1/4,2/3)=3/4x-1/2y$ and ${U}_{2}(y,x,1/4,2/3)=1/3x-1/3y$. Thus, $arg{max}_{x}{U}_{2}(y,x,1/4,2/3)=\left\{3y\right\}$. Put

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Krishna, V.; Perry, M. Efficient Mechanism Design; Manuscript; Department of Economics, The Pennsylvania State University: University Park, PA, USA, 1998. [Google Scholar]
- Lucier, B.; Borodin, A. Price of anarchy for greedy auctions. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, TX, USA, 17–19 January 2010; SIAM: Philadelphia, PA, USA, 2010; pp. 537–553. [Google Scholar]
- Hartline, J.D. Bayesian mechanism design. Found. Trends Theor. Comput. Sci.
**2013**, 8, 143–263. [Google Scholar] [CrossRef] - Hartline, J.D. Mechanism Design and Approximation Manuscript. 2017. Available online: http://jasonhartline.com/MDnA/ (accessed on 6 April 2018).
- Rabinovich, Z.; Naroditskiy, V.; Gerding, E.H.; Jennings, N.R. Computing pure bayesian-nash equilibria in games with finite actions and continuous types. Artif. Intell.
**2013**, 195, 106–139. [Google Scholar] [CrossRef] - Edalat, A.; Ghoroghi, A.; Sakellariou, G. Multi-games and a double game extension of the prisoner’s dilemma. arXiv, 2012; arXiv:1205.4973. [Google Scholar]
- Ghoroghi, A. Multi-Games and Bayesian Nash Equilibriums. Ph.D. Thesis, Department of Computing, Imperial Collage London, London, UK, 2015. [Google Scholar]
- Harris, M.; Townsend, R.M. Resource allocation under asymmetric information. Econom. J. Econom. Soc.
**1981**, 49, 33–64. [Google Scholar] [CrossRef] - Holmström, B.; Myerson, R.B. Efficient and durable decision rules with incomplete information. Econom. J. Econom. Soc.
**1983**, 51, 1799–1819. [Google Scholar] [CrossRef] - Crémer, J.; McLean, R.P. Optimal selling strategies under uncertainty for a discriminating monopolist when demands are interdependent. Econom. J. Econom. Soc.
**1985**, 53, 345–361. [Google Scholar] [CrossRef] - Bergemann, D.; Morris, S. Ex post implementation. Games Econ. Behav.
**2008**, 63, 527–566. [Google Scholar] [CrossRef] - Bulow, J.I.; Geanakoplos, J.D.; Klemperer, P.D. Multimarket oligopoly: Strategic substitutes and complements. J. Political Econ.
**1985**, 93, 488–511. [Google Scholar] [CrossRef] - Abolhassani, M.; Bateni, M.H.; Hajiaghayi, M.; Mahini, H.; Sawant, A. Network cournot competition. In Proceedings of the International Conference on Web and Internet Economics, Beijing, China, 14–17 December 2014; Springer: Cham, Switzerland, 2014; pp. 15–29. [Google Scholar]
- Bimpikis, K.; Ehsani, S.; Ilkilic, R. Cournot competition in networked markets. In Proceedings of the Fifteenth ACM Conference on Economics and Computation, Palo Alto, CA, USA, 8–12 June 2014; p. 733. [Google Scholar]
- Yanovskaya, E.B. Equilibrium points in polymatrix games. Litov. Mat. Sb.
**1968**, 8, 381–384. [Google Scholar] - Kearns, M.; Littman, M.L.; Singh, S. Graphical models for game theory. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, Seattle, WA, USA, 2–5 August 2001; Morgan Kaufmann Publishers Inc.: San Francisco, CA, USA, 2001; pp. 253–260. [Google Scholar]
- Papadimitriou, C.H.; Roughgarden, T. Computing correlated equilibria in multi-player games. J. ACM (JACM)
**2008**, 55, 14. [Google Scholar] [CrossRef] - Ortiz, L.E.; Irfan, M.T. Tractable algorithms for approximate Nash equilibria in generalized graphical games with tree structure. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, San Francisco, CA, USA, 4–9 February 2017; pp. 635–641. [Google Scholar]
- Howson, J.T., Jr.; Rosenthal, R.W. Bayesian equilibria of finite two-person games with incomplete information. Manag. Sci.
**1974**, 21, 313–315. [Google Scholar] [CrossRef] - Austrin, P.; Braverman, M.; Chlamtác, E. Inapproximability of NP-complete variants of Nash equilibrium. Theory Comput.
**2013**, 9, 117–142. [Google Scholar] [CrossRef] - Rubinstein, A. Inapproximability of Nash equilibrium. In Proceedings of the Forty-Seventh Annual ACM Symposium On Theory of Computing, Portland, OR, USA, 14–17 June 2015; ACM: New York, NY, USA, 2015; pp. 409–418. [Google Scholar]
- Baumann, L. A Model of Weighted Network Formation. 2017. Available online: https://ssrn.com/abstract=2533533 (accessed on 16 July 2018).
- Damasio, A.; Tranel, D.; Damasio, H. Somatic markers and the guidance of behavior: Theory and preliminary testing. In Frontal Lobe Function and Dysfunction; Levin, H.S., Eisenberg, H.M., Benton, A.L., Eds.; Oxford University Press: New York, NY, USA, 1991. [Google Scholar]
- Bechara, A.; Damasio, H.; Damasio, A.R. Emotion, decision making and the orbitofrontal cortex. Cereb. Cortex
**2000**, 10, 295–307. [Google Scholar] [CrossRef] [PubMed] - Loewenstein, G.; Lerner, J.S. The role of affect in decision making. Handb. Affect. Sci.
**2003**, 619, 3. [Google Scholar] - Gintis, H. The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences; Princeton University Press: Princeton, NJ, USA, 2009. [Google Scholar]
- Ostrom, E. Biography of Robert Axelrod. PS Political Sci. Politics
**2007**, 40, 171–174. [Google Scholar] [CrossRef] - Shubik, M. Game theory, behavior, and the paradox of the Prisoner’s Dilemma: Three solutions. J. Confl. Resolut.
**1970**, 14, 181–193. [Google Scholar] [CrossRef] - Hausman, D.M. Taking the prisoner’s dilemma seriously: What can we learn from a trivial game? In The Prisoner’s Dilemma; Peterson, M., Ed.; Cambridge University Press: Cambridge, UK, 2015. [Google Scholar]
- Khadjavi, M.; Lange, A. Prisoners and their dilemma. J. Econ. Behav. Organ.
**2013**, 92, 163–175. [Google Scholar] [CrossRef] - Brosig, J. Identifying cooperative behavior: Some experimental results in a Prisoner’s Dilemma game. J. Econom. Behav. Organ.
**2002**, 47, 275–290. [Google Scholar] [CrossRef] - Berg, J.; Dickhaut, J.; McCabe, K. Trust, reciprocity, and social history. Games Econ. Behav.
**1995**, 10, 122–142. [Google Scholar] [CrossRef] - Chaudhuri, A.; Gangadharan, L. An experimental analysis of trust and trustworthiness. South. Econ. J.
**2007**, 73, 959–985. [Google Scholar] - Johnson, N.D.; Mislin, A.A. Trust games: A meta-analysis. J. Econ. Psychol.
**2011**, 32, 865–889. [Google Scholar] [CrossRef] - Davis, J.B. Individuals and Identity in Economics; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Fudenberg, D.; Tirole, J. Game Theory; The MIT Press: Cambridge, MA, USA, 1991. [Google Scholar]
- Devanur, N.; Hartline, J.D.; Karlin, A.; Nguyen, T. Prior-independent multi-parameter mechanism design. In Proceedings of the International Workshop on Internet and Network Economics, Singapore, 11–14 December 2011; Springer: Berlin/Heidelberg, Germany, 2011; pp. 122–133. [Google Scholar]
- Fu, H.; Hartline, J.D.; Hoy, D. Prior-independent auctions for risk-averse agents. In Proceedings of the Fourteenth ACM Conference on Electronic Commerce, Philadelphia, PA, USA, 16–20 June 2013; ACM: New York, NY, USA, 2013; pp. 471–488. [Google Scholar]
- Chen, P.-A.; De Keijzer, B.; Kempe, D.; Schäfer, G. The robust price of anarchy of altruistic games. In Proceedings of the International Workshop on Internet and Network Economics, Singapore, 11–14 December 2011; Springer: Berlin/Heidelberg, Germany, 2011; pp. 383–390. [Google Scholar]
- Axelrod, R.M. The Evolution of Cooperation; Basic Books: New York, NY, USA, 2006. [Google Scholar]
- Ounsley, J. The Prisoner’s Dilemma and Our Morals. Master’s Thesis, Department of Computing, Imperial Collage London, London, UK, 2010. [Google Scholar]
- McKelvey, R.D.; McLennan, A. Computation of equilibria in finite games. Handb. Comput. Econ.
**1996**, 1, 87–142. [Google Scholar]

**Figure 1.**NEs in local games of three double games for PD, the shaded regions represent the continuous types: (

**a**) ${\mathsf{\Theta}}_{1}={\mathsf{\Theta}}_{2}=[0,1]$, $\mu <\lambda $, with no pure ex post NE; (

**b**) ${\mathsf{\Theta}}_{1}={\mathsf{\Theta}}_{2}=[0,\mu ]\cup [\lambda ,1]$, $\mu <\lambda $, with one pure ex post NE; and (

**c**) ${\mathsf{\Theta}}_{1}={\mathsf{\Theta}}_{2}=[0,1]$, $\mu =\lambda $ with four pure ex post NEs.

**Figure 3.**The partition of the type space of Agent 2 for each extreme type of Agent 1 in the multigame of Example 4. The multigame has no compatible pure ex post NE by Theorem 2. The type ${\theta}_{2}=(1/4,1/4,1/2)$ shows that the partition of ${\mathsf{\Theta}}_{2}$ is not independent of ${\theta}_{1}\in {\mathsf{\Theta}}_{1}^{e}$.

**Figure 4.**The partitioning of ${\mathsf{\Theta}}^{b}$ in a two-agent multigame with three games which shows that the multigame has a pure ex post NE. The types of each agent in the three partitioning of its type space are shown in three different colours.

C | D | C | D | ||
---|---|---|---|---|---|

C | $(r,r)$ | $(s,t)$ | C | $(y,y)$ | $(y,z)$ |

D | $(t,s)$ | $(p,p)$ | D | $(z,y)$ | $(z,z)$ |

C | D | C | D | ||
---|---|---|---|---|---|

C | (6, 6) | (1, 7) | C | (5, 5) | (5, 1) |

D | (7, 1) | (2, 2) | D | (1, 5) | (1, 1) |

C | D | C | D | ||
---|---|---|---|---|---|

C | (16, 16) | (3, 20) | C | (15, 15) | (15, 3) |

D | (20, 3) | (6, 6) | D | (3, 15) | (3, 3) |

u | v | u | v | ||
---|---|---|---|---|---|

s | $({b}_{1},{b}_{2})$ | $({d}_{1},{d}_{2})$ | s | $({g}_{1},{g}_{2})$ | $({h}_{1},{h}_{2})$ |

t | $({e}_{1},{e}_{2})$ | $({f}_{1},{f}_{2})$ | t | $({k}_{1},{k}_{2})$ | $({\ell}_{1},{\ell}_{2})$ |

a = u_{3} | a = v_{3} | ||||
---|---|---|---|---|---|

u_{2} | v_{2} | u_{2} | v_{2} | ||

${u}_{1}$ | $(1,10,3)$ | $(6,9,9)$ | ${u}_{1}$ | $(12,3,0)$ | $(3,1,6)$ |

${v}_{1}$ | $(0,11,11)$ | $(4,7,10)$ | ${v}_{1}$ | $(9,13,8)$ | $(-1,12,4)$ |

a = u_{3} | a = v_{3} | ||||
---|---|---|---|---|---|

u_{2} | v_{2} | u_{2} | v_{2} | ||

${u}_{1}$ | $(1,3,1)$ | $(8,6,4)$ | ${u}_{1}$ | $(5,2,2)$ | $(7,8,5)$ |

${v}_{1}$ | $(2,8,12)$ | $(10,20,18)$ | ${v}_{1}$ | $(8,2,13)$ | $(11,5,20)$ |

${\mathit{u}}_{2}$ | ${\mathit{v}}_{2}$ | ${\mathit{w}}_{2}$ | ${\mathit{u}}_{2}$ | ${\mathit{v}}_{2}$ | ${\mathit{w}}_{2}$ | ${\mathit{u}}_{2}$ | ${\mathit{v}}_{2}$ | ${\mathit{w}}_{2}$ | |||
---|---|---|---|---|---|---|---|---|---|---|---|

${u}_{1}$ | $(3,3)$ | $(3,2)$ | $(3,2)$ | ${u}_{1}$ | $(2,2)$ | $(2,3)$ | $(2,2)$ | ${u}_{1}$ | $(2,2)$ | $(2,2)$ | $(1,3)$ |

${v}_{1}$ | $(2,3)$ | $(2,2.5)$ | $(2,2)$ | ${v}_{1}$ | $(3,2)$ | $(3,3)$ | $(3,1)$ | ${v}_{1}$ | $(2,2)$ | $(2,2)$ | $(2,3)$ |

${w}_{1}$ | $(2,3)$ | $(2,2)$ | $(2,1)$ | ${w}_{1}$ | $(2,2)$ | $(2,3)$ | $(2,1)$ | ${w}_{1}$ | $(3,2)$ | $(3,1)$ | $(3,3)$ |

**Table 8.**NE of ${G}_{({\theta}_{1},{\theta}_{2})}$ for all $({\theta}_{1},{\theta}_{2})\in {\mathsf{\Theta}}^{e}$.

${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{3}$ | |
---|---|---|---|

${p}_{1}$ | $({u}_{1},{u}_{2})$ | $({u}_{1},{v}_{2})$ | $({u}_{1},{w}_{2})$ |

${p}_{2}$ | $({v}_{1},{u}_{2})$ | $({v}_{1},{v}_{2})$ | $({v}_{1},{w}_{2})$ |

${p}_{3}$ | $({w}_{1},{u}_{2})$ | $({w}_{1},{v}_{2})$ | $({w}_{1},{w}_{2})$ |

a | r | a | r | ||
---|---|---|---|---|---|

a | $({b}_{1},{b}_{2})$ | $({d}_{1},{d}_{2})$ | a | $({g}_{1},{g}_{2})$ | $({h}_{1},{h}_{2})$ |

r | $({e}_{1},{e}_{2})$ | $({f}_{1},{f}_{2})$ | r | $({k}_{1},{k}_{2})$ | $({\ell}_{1},{\ell}_{2})$ |

${\mathit{s}}_{21}$ | ${\mathit{s}}_{22}$ | ${\mathit{s}}_{23}$ | ${\mathit{s}}_{21}$ | ${\mathit{s}}_{22}$ | ${\mathit{s}}_{23}$ | ${\mathit{s}}_{21}$ | ${\mathit{s}}_{22}$ | ${\mathit{s}}_{23}$ | |||
---|---|---|---|---|---|---|---|---|---|---|---|

${s}_{11}$ | $(3,7)$ | $(11,4)$ | $(4,3)$ | ${s}_{11}$ | $(4,1)$ | $(5,4)$ | $(9,2)$ | ${s}_{11}$ | $(2,2)$ | $(3,6)$ | $(7,10)$ |

${s}_{12}$ | $(2,8)$ | $(10,5)$ | $(3,4)$ | ${s}_{12}$ | $(5,5)$ | $(6,8)$ | $(10,6)$ | ${s}_{12}$ | $(4,4)$ | $(6,8)$ | $(8,12)$ |

${s}_{13}$ | $(1,5)$ | $(9,2)$ | $(2,1)$ | ${s}_{13}$ | $(3,0)$ | $(4,3)$ | $(8,1)$ | ${s}_{13}$ | $(3,6)$ | $(5,10)$ | $(9,14)$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Edalat, A.; Hossein Ghorban, S.; Ghoroghi, A. Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments. *Games* **2018**, *9*, 85.
https://doi.org/10.3390/g9040085

**AMA Style**

Edalat A, Hossein Ghorban S, Ghoroghi A. Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments. *Games*. 2018; 9(4):85.
https://doi.org/10.3390/g9040085

**Chicago/Turabian Style**

Edalat, Abbas, Samira Hossein Ghorban, and Ali Ghoroghi. 2018. "Ex Post Nash Equilibrium in Linear Bayesian Games for Decision Making in Multi-Environments" *Games* 9, no. 4: 85.
https://doi.org/10.3390/g9040085