each player i knows \\(u_{i}(a_{i}, a_{j})\\) and \\(u_{j}(a_{i}, a_{j})\\) for every \\((a_{i}, a_{j})\\)<\/p>\n\n\n\n
However, this assumption is not in general satisfied: players are not<\/em> completely informed about the payoffs in most situations. Then we say information is incomplete<\/em>. In the remaining two parts, we relax the assumption of (CI<\/mark><\/strong>) and study how to analyze the games.<\/p>\n\n\n\n One example of incomplete information is the next:<\/p>\n\n\n\n EXAMPLE 4.1 (FIRST-PRICE AUCTIONS)<\/mark>. Suppose an indivisible goods is auctioned to two bidders. Each bidder i knows his own valuation \\(\\nu_{i}\\) but does not know \\(\\nu_{j}\\). For each combination of bids \\((b_{1}, b_{2})\\), bidder 1 knows his own payoff<\/p>\n\n\n\n \\(u_{1}(b_{1}, b_{2})\\) = \\(\\begin{cases} & \\nu_{1} – b_{1} \\text{ if } b_{1} > b_{2}\\\\ & \\text{ 0 otherwise, } \\end{cases}\\)<\/p>\n\n\n\n but is uncertain about bidder 2’s payoff<\/p>\n\n\n\n \\(u_{2}(b_{1}, b_{2})\\) = \\(\\begin{cases} & \\nu_{2} – b_{1} \\text{ if } b_{2} > b_{1}\\\\ & \\text{ 0 otherwise, } \\end{cases}\\)<\/p>\n\n\n\n because \\(\\nu_{2}\\)<\/mark><\/strong> is bidder 2’s private information.<\/p>\n\n\n\n Incomplete information pervades a variety of (almost all) economic environments:<\/p>\n\n\n\n (1) bargaining<\/em>: each party does not know the other’s willingness-to-pay<\/p>\n\n\n\n B: Buyer S: Seller (2) market competition<\/em>: firms do not know the cost of their competitiors<\/p>\n\n\n\n \\(u_{1}(q_{1}, q_{2}) = p \\cdot q_{1} – c_{1}(q_{1}), p \\cdot q_{2} – c_{2}(q_{2})\\)<\/mark><\/p>\n\n\n\n (3) contracting<\/em>: employers do not know he level of skills of employees<\/p>\n\n\n\n \\(\\pi\\): company’s profit (4) public good provision<\/em>: the planner does not know the exact value of the public good to a society<\/p>\n\n\n\n \\(w = \\sum_{i=1}^{n}\\nu_{i} – c\\)<\/mark><\/p>\n\n\n\n For now on, we tackle on incomplete information<\/p>\n\n\n\n As before, we subdivide the class of games into two:<\/p>\n\n\n\n (1) static<\/em> games with incomplete info, or (static) Bayesian games, and The solution concept of (1) and (2) is a generalization of Nash eqbm and SPE, respectively<\/p>\n\n\n\n 4. Bayesian Games We start with one example that formalizes the key idea<\/em> of Bayesian games:<\/p>\n\n\n\n EXAMPLE 4.2 (A GUESSING GAME, EXERCISE 2 ON PAGE 333)<\/mark>. Suppose Andy and Brian play a guessing game<\/p>\n\n\n\n Question<\/mark>: How to model Andy’s private information (or type)?<\/p>\n\n\n\n The idea of JOHN HARSANYI<\/mark><\/strong> (1967, 1968) is to introduce a move by Nature<\/em>, which transforms<\/p>\n\n\n\n a problem of incomplete<\/em> info into the problem of imperfect<\/em> info<\/p>\n\n\n Figure: The extensive-form representation of Example 4.2<\/p>\n\n\n\n In this example, Nature’s move is observed only by Andy (= player 1).<\/p>\n\n\n\n EXAMPLE 4.3 (THE GIFT GAME ON PAGE 327)<\/mark>.<\/p>\n\n\n Observe that Nature’s move results in two different<\/mark><\/strong> info sets. P1’s strategy therefore must specify what he intends to do if he is a friend and if he is an enemy.<\/p>\n\n\n\n In this sense, a player’s strategy is type<\/mark><\/strong>-contingent in Bayesian games<\/p>\n\n\n\n In the example, the players’ possible strategies are<\/p>\n\n\n\n \\(S_{1} = {N^{F}N^{E}, G^{F}G^{E}, N^{F}G^{E}, G^{F}N^{E}}\\) and \\(S_{2}\\) = {A, R}<\/p>\n\n\n\n It is an incomplete information game, but we can change it to imperfect information through the harsanyi idea.<\/mark> Notice that in the gift game, both SPE and NE give us the same prediction<\/p>\n\n\n\n To find an equilibrium, we first put the game in the normal form<\/p>\n\n\n\n The procedure is the same as before, except<\/p>\n\n\n\n \\((s_{1}, s_{2})\\) = one path w\/o Nature’s move but \\((s_{1}, s_{2})\\) \u2260<\/mark><\/strong> one path with Nature’s move<\/p>\n\n\n\n As a result, a strategy profile \\((s_{1}, s_{2})\\) does not pin down players’ payoff<\/p>\n\n\n\n The payoffs are averaged<\/em> according to Nature’s probability distribution<\/p>\n\n\n\n For example, \\((N^{F}G^{E}, A)\\): Depending on Nature’s move, the players obtain either (0, 0) or (1, -1)<\/p>\n\n\n\n The expected payoff<\/mark><\/strong> from \\((N^{F}G^{E}, A)\\) is<\/p>\n\n\n\n \\(u_{1}\\) = 1\u2219(1-p) = 1-p EXAMPLE 4.4<\/mark>.<\/p>\n\n\n\n x = \\(\\begin{cases} & \\text{ 12 with probability 2\/3 } \\\\ & \\text{ 0 with probability 1\/3 } \\end{cases}\\)<\/p>\n\n\n\n Players 1 and 2 play the above normal-form game<\/p>\n\n\n\n The realization of x is player 1’s private information<\/mark><\/strong><\/p>\n\n\n\n Player 2 knows only that x = 12 with probability 2\/3 and x = 0 with probability 1\/3<\/p>\n\n\n Nature’s move in this example determines which normal-form game is to be played<\/p>\n\n\n\n <\/p>\n\n\n\nIntroduction<\/h3>\n\n\n\n
\\(u_{B}(P) = v_{B} – P, u_{s}(P) = P – v_{s}\\)<\/mark><\/p>\n\n\n\n
\\(\\pi(w, hire) = \\nu (e, t) – w)\\)<\/mark><\/p>\n\n\n\n
(2) dynamic<\/em> games with incomplete info, or dynamic Bayesian games<\/p>\n\n\n\ninfo \\ player’s moves<\/td> simultaneous<\/td> sequential<\/td><\/tr> complete<\/td> Part I: Normal-Form Games
ISD, Nash Eqbm<\/strong><\/td>Part II: Extensive-Form Games
Subgame Perfect Eqbm<\/strong><\/td><\/tr>incomplete<\/td> Part III: Static Bayesian Games
Bayesian Nash Eqbm<\/strong><\/td>Part IV: Dynamic Bayesian Games
Perfect Bayesian Eqbm<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
4.1 Random Events and Incomplete Information (Chapter 24)<\/p>\n\n\n\n\n
![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/06\/image-2-1024x893.png\")
<\/figure><\/div>\n\n\n![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/06\/image-3-1024x677.png\")
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There are two equilibrium notions we have learned so far: NE and SPE.<\/mark>
Since there is no proper subgame, the prediction of NE and SPE is the same.<\/mark><\/p>\n\n\n\nBayesian Normal Form<\/h3>\n\n\n\n
1 \\ 2<\/td> A<\/td> R<\/td><\/tr> \\(G^{F}G^{E}\\)<\/td> 1, 2p-1<\/td> -1, 0<\/td><\/tr> \\(G^{F}N^{E}\\)<\/td> p, p<\/td> -p, 0<\/td><\/tr> \\(N^{F}G^{E}\\)<\/td> 1-p, p-1<\/td> p-1, 0<\/td><\/tr> \\(N^{F}N^{E}\\)<\/td> 0, 0<\/td> 0, 0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\\(u_{2}\\) = -1\u2219(1-p) = p-1<\/mark><\/p>\n\n\n\n1 \\ 2<\/td> C<\/td> D<\/td><\/tr> A<\/td> x, 9<\/td> 3, 6<\/td><\/tr> B<\/td> 6, 0<\/td> 6, 9<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n ![\"\"](\"https:\/\/saraheee.com\/wp-content\/uploads\/2023\/06\/image-4-1024x912.png\")
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