# bayesian game

• The second is called the induced normal form (see Section 6.3.3 of Multiagent Systems[10]) which still has players yet expands the number of each player i’s actions from to
, i.e., the pure policy is a combination of actions the player should take for different types.

• Player1 will never have complete information about player2, but may be able to infer the probability of type1 and type2 appearing from whether the previous firm entering the
market was blocked, it is a Bayesian game.

• In a non-Bayesian game, a strategy profile is a Nash equilibrium if every strategy in that profile is a best response to every other strategy in the profile; i.e., there is
no strategy that a player could play that would yield a higher payoff, given all the strategies played by the other players.

• Roughly speaking, Harsanyi defined Bayesian games in the following way: players are assigned by nature at the start of the game a set of characteristics.

• Therefore, players can be essentially modelled as having incomplete information and the probability space of the game still follows the law of total probability.

• In game theory, a Bayesian game is a strategic decision-making model which assumes players have incomplete information.

• Thus, the payoff matrix of this Normal-form game for both players depends on the type of the suspect.

• The first is called the agent-form game (see Theorem 9.51 of the Game Theory book[9]) which expands the number of players from to , i.e., every type of each player becomes
a player.

• An analogous concept can be defined for a Bayesian game, the difference being that every player’s strategy maximizes their expected payoff given their beliefs about the state
of nature.

• ad infinitum – common knowledge), play in the game will be as follows according to perfect Bayesian equilibrium:[25][26] When the type is “criminal”, the dominant strategy
for the suspect is to shoot, and when the type is “civilian”, the dominant strategy for the suspect is not to shoot; alternative strictly dominated strategy can thus be removed.

• An information set of player i is a subset of player i’s decision nodes that she cannot distinguish between.

• They are notable because they allowed, for the first time in game theory, for the specification of the solutions to games with incomplete information.

• Pure strategies In a strategic game, a pure strategy is a player’s choice of action at each point where the player must make a decision.

• That is, a strategy profile is a Bayesian Nash equilibrium if and only if for every player keeping the strategies of every other player fixed, strategy maximizes the expected
payoff of player according to that player’s beliefs.

• That is, if player i is at one of her decision nodes in an information set, she does not know which node within the information set she is at.

• This game is defined by (N,A,T,p,u), where: If both players are rational and both know that both players are rational and everything that is known by any player is known to
be known by every player (i.e.

• A player recognises payoffs as expected values based on a prior distribution of all possible types.

• [23] Another approach is to assume that players within any collective agent know that the agent exists, but that other players do not know this, although they suspect it with
some probability.

• Players know their own type, but only a probability distribution of other players.

• There is a probability p that the suspect is a criminal, and a probability 1-p that the suspect is a civilian; both players are aware of this probability (common prior assumption,
which can be converted into a complete-information game with imperfect information).

• [24] For example, Alice and Bob may sometimes optimize as individuals and sometimes collude as a team, depending on the state of nature, but other players may not know which
of these is the case.

• If players do not have private information, the probability distribution over types is known as a common prior.

• Player 1 does not and believes that the value v of the car to the owner (Player 2) is distributed uniformly between 0 and 100 (i.e., each of two value sub-intervals of [0,
100] of equal length are equally likely).

• • Consider two players with a zero-sum objective function.

• A Bayesian Nash equilibrium (BNE) is defined as a strategy profile that maximizes the expected payoff for each player given their beliefs and given the strategies played by
the other players.

• By mapping probability distributions to these characteristics and by calculating the outcome of the game using Bayesian probability, the result is a game whose solution is,
for technical reasons, far easier to calculate than a similar game in a non-Bayesian context.

• Except, by attaching probability to the game, the final game functions as though it were an incomplete information game.

• • Only “lemons” (used cars in bad conditions, specifically with value at most equal to p) are traded • Player 1 can guarantee herself a payoff of zero by bidding 0, hence
in equilibrium, • Since only “lemons” (used cars in bad conditions) are traded, the market collapses • No trade is possible even when trade would be economically efficient[27] Enter the monopolized market A new company (player1) that
wants to enter a market that is monopolised by a large company will encounter two types of monopolist (player2), type1 is prevented and type2 is allowed.

• A player’s beliefs about the state of nature are formed by conditioning the prior probabilities on the player’s own type according to Bayes’ rule.

• One approach is to continue to treat individual players as reasoning in isolation, but to allow them, with some probability, to reason from the perspective of a collective.

• Players hold private information relevant to the game, meaning that the payoffs are not common knowledge.

Works Cited

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