The priacy paradox of Rann Smorodinsky

[Gil Kalai]

Dear DT forumairs

The following privacy paradox due to Rann Smorodinsky is discussed over my blog.

http://gilkalai.wordpress.com/2012/12/05/the-privacy-paradox-of-rann-smorodinsky/

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Suppose that you have the following one-time scenario. You want to buy a sandwich where the options are a roast beef sandwich or an avocado sandwich. Choosing the sandwich of your preference (say, the avocado sandwich) adds one to your utility, but having your private preference known to the seller,  reduces by one your utility. The prior people have on your preferences is fifty-fifty.

If you choose the avocado sandwich your utility is zero, hence you can improve on this by picking each type of sandwich at random with probability 1/2. In this case your private preference remains unknown and you gain in expectation 1/2 for having the sandwich you prefer with probability 1/2.

But given this strategy, you can still improve on it by choosing the avocado sandwich.

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As always, comments will be highly appreciated.

best regards, Gil

--
Gil Kalai


Phone: +972-2-6584729
e-mail: gil....@gmail.com
blog: "Combinatorics and More"  http://gilkalai.wordpress.com/
home page http://www.ma.huji.ac.il/~kalai/

[Ken Binmore]

I don't see any paradox since the mixed strategy solution depends on the seller knowing that you are using a mixed strategy.

              ken Binmore

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[Spiegler, Ran]

This situation can (should?) be modeled as a psychological game, after Geanakopolos, Pearce and Stachetti 1989. In such a game, players' payoffs directly depend on their (and their opponents') hierarchy of beliefs. In Rann's example, the buyer's payoff can be defined as the value of the product he ends up buying minus 2*|p-0.5|, where p is the seller's posterior on avocado. Equilibrium in such a game is defined more or less as usual (in general, since payoffs can depend on high order beliefs, the equilibrium concept has to deal with those, too). When the buyer orders avocado for sure, this is an equilibrium. It does not maximize the buyer's ex ante expected payoffs, but that is not unusual in this class of games because payoffs need not be linear in beliefs.

Here is another example, which Kfir Eliaz and I used to call "the dress game". Bob's girlfriend bought a dress. He observes whether it is nice. His strategy is to tell her whether it is nice, given his observation. His objective is to maximize a convex increasing function of his girfriend's posterior that the dress is nice. TE only equilibria are babbling equilibria, in which Bob's message is totally uninformative, because otherwise he always has an incentive to deviate from his equilibrium (potentially mixed) strategy and say the dress is nice. By the convexity assumption, the babbling equilibria do not maximize Bob's ex ante expected payoff. If he could commit, he would want to be fully informative.

 

Ran Spiegler

School of Economics, Tel Aviv University

&

Department of Economics, University College London

 

URL: http://www.tau.ac.il/~rani

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From: decision_t...@googlegroups.com [decision_t...@googlegroups.com] on behalf of Ken Binmore [k.bi...@ucl.ac.uk]
Sent: 06 December 2012 12:16
To: decision_t...@googlegroups.com
Subject: Re: [DT_Forum] The priacy paradox of Rann Smorodinsky

[Ambrus, Attila]

Just saw a new paper, by Ely, Frankel and Kamenica, at a conference that is related to these kind of games. They do assume ex ante commitment:

http://faculty.chicagobooth.edu/emir.kamenica/documents/suspense.pdf

For a “paradox” without commitment, there’s also the Hangman’s paradox by John Geanakoplos.

Best, Attila

[tzachi...@gmail.com]

Indeed, psychological games come to mind.  But one could still argue that there is room for a paradox: back in '88 David (Schmeidler) and I had a paper titled "Information-Dependent Games", in which one's payoff depended on beliefs, and where we used the surprise-test paradox to generate a trivial example where no equilibrium existed.  

The main difference between GPS and our paper was, that GPS required linearity in probabilities.  In some sense, they allowed the players to use more strategies than the others' beliefs could depend on.  I think that if the payoff could freely (not-necessarily linearly) depend on mixed strategies, the non-existence could be established in their model as well.

Best,

Tzachi

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From: "Spiegler, Ran" <r.spi...@ucl.ac.uk>

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Date: Thu, 6 Dec 2012 15:46:47 +0000

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Subject: RE: [DT_Forum] The priacy paradox of Rann Smorodinsky

[Tymofiy Mylovanov]

A similar argument is used to rule out undesired equilibria in (informed principal) games in which a privately informed player proposes a bargaining contract. 

A principal and an agent each own half of the good. They have independent private values. For simplicity, assume that the principal is equally likely to value the good at either 0 or 1 and that the agent's valuation is uniform on [0,1]. The principal chooses a bargaining contract. If the agent agrees, they play according to the contract. Otherwise, they keep the shares. 

To rule out undesirable equilibria, one considers the following deviation. The principal asks the agent to name a price of either 0 or 1 and then decides whether to sell her share or buy the agent's share at this price. (He might need to pay epsilon to the agent to induce him to accept this mechanism.) 

If the agent chooses price 0, type 1 of the principal will find it profitable to deviate to this mechanism and the agent will sell at a loss. If the agent chooses price 1, type 0 will deviate to this mechanism and the agent will buy at a loss. The agent breaks even if and only if he mixes 50/50 between both prices. All other mixed strategies will generate a loss. 

However, in any incentive compatible and individually rational mechanism, the agent's expected payoff is strictly positive. 

Tymofiy Mylovanov
----------------------------------------------------
Department of Economics
University of Pennsylvania
3718 Locust Walk
Philadelphia, PA 19104

https://sites.google.com/site/tmylovanov/
814-321-7744

[g charles-cadogan]

My comments on the game focuses on the hypothesized probability distribution from a prospect theory perspective.  One of the empirical regularities of prospect theory is that people hate symmetric 50-50 bets. Based on the information given, the introduction of a gain utility (+1) for avocado (“A”), and a loss utility (-1) for roast beef (“RB” phonetic Arby),  imply a reference point of 0 against which gain and loss is measured. If so, then let p be the probability associated with selecting A, and q that with selecting RB. The “mixed prospect” is now (A, p; 0, 1-p-q ; RB, q). One of the empirical regularities of probability weighting functions is that they have a fixed point probability of approximately 1/3 that separates loss and gain domains based on ranking. In which case an admissible criterion is that 1-p-q=1/3 So that p+q=2/3. Since the “rank dependent utility”  implied by the problem is A > 0 > RB, we have the weighting scheme w(p), w(1/3),  w(q). So that the introduction of a prior probability of p=1/2 è q=1/6 In which case w(1/6) > 1/6 due to overweighting  in“loss probability” domain; w(1/2) < 1/2 due to underweighting gain probability domain; and w(1/3)=1/3 due to fixed point.  Roughly, the decision weights for gain (g) and loss (l) respectively, are pi_g=w^+(1/3 + 1/2) – w^+(1/3) and pi_l =w^-(1/6 + 1/3) – w^-(1/6). For a value function V the nonexpected value of the choices available to our subject is V(.)=V^+(+1)pi_g + V^-1(-1)pi_l. Evidently, the nonexpected utility is also greater than zero even if the seller knows the subject will select avocado. For  admissible “strategies” that “dominate” ½ we can impose the constraint  0 < V^+(+1)pi_g + V^-1(-1)pi_l > 1/2. So that any combination of value function and decision weights that support that inequality would be “strategy proof” in so far as whether the seller knows the subject selects avocado or not.

[Gil Kalai]

I wonder if the following strategy to protect your true private preferences between the sandwiches works: You come to the seller and tell him: I was brought up that beef sandwich goes with red wine, and avocado sandwich goes with beer. And I really much prefer beer!  --Gil

[Gil Kalai]

Dear DT forumairs

I want to draw your attention to my blog post (clicking will take you there)

Riddles (Stumpers), Psychology, and AI.

Six years ago  Maya Bar-Hillel, Tom Noah and Shane Frederick wrote a paper, Learning psychology from riddles: The case of stumpers, and three years ago Maya wrote another paper 

An annotated compendium of stumpers where people's difficulties in solving various simple riddles (called stumpers) is associated with works of Eleanor Rosch and Paul Grice. The post describes our joint project on testing these riddles with AI. 

Here is one such riddle.

A ping pong ball was hit. It flew in the air, stopped, reversed direction, and returned to where it originated. The ball was not attached to anything, nor bounced off anything. Explain in a few words how that is possible.

best regards Gil Kalai