Denis, your story strikes a Romanian chord. The situation around here is even worse, from what I can tell. But it is quite a fascinating question, with different answers from different points of view.
For an economist, it is a matter of price formation. In the state system, Romanian doctors are paid a fixed (and miserable) wage, largely unrelated to quality or effort. The incentive to pocket bribes is huge, and patients know it so well. In the private sector (with transparent and varied prices for medical services), bribes are almost unheard of. Also, there is a more or less efficient market for bribes. Patients find out how much a doctor expects, usually from past patients, or from other doctors. Surgeons receive more than GPs, professors more than debutants, etc.
But I think there is something more about "medical envelopes", from a cognitive point of view. First of all, there is a vast asymmetry of competence between doctors and patients, which gives the former a large freedom of action. Is this pill better, or another one? Surgery or not? Home treatment or hospitalisation? To make things worse, the post-hoc reckoning is not very helpful, since most decisions may be medically justified, but you might also end up dead. The patient is at the mercy of the practitioner since she does not know what choices are better. The best way to make sure one gets the proper treatment is to insure the benevolence of the doctor, and a bribe is the simplest path to gain the doctor's amity.
Second, there is something special about this particular social exchange: the patient is dealing in an ultimate value - her health. Something everyone in Romania says is that there is no price too high to be healthy. (Paradoxically, giving up smoking somehow does not make the list - self-hint-hint-nudge-nudge). If people would risk not bribing a policeman to avoid a fine, they are extremely unlikely to jeopardise their health in this manner. One cannot afford to stick to abstract principles (like discouraging corruption) when her life is at stake.
Finally, there is something like a Maussian gift in the affair: one passes a fat envelope even without the explicit mention of an economic exchange. It is not that the surgeon would not operate without being bribed - the patient just shows gratitude without visible economic reckoning. Of course, under the veil of generosity stands the solid self-interest of the patient. The fat envelope is meant to make sure that no scalpel is lost in her belly. But no-one says it out loud. It's a "I know that you know that I know etc" which makes sure that the transaction is smooth and polite.
To end with a personal anecdote: I was (and to some extent I still am) very wary of giving out envelopes to doctors. A little bit of moral prudishness, a little bit of fear (what if he feels insulted?), a bit of monetary unsaviness. Those who are more competent in these matters reassured me: "just put the envelope on his desk - he knows what to do next" After all, he is the expert, and I am not.
It looks like managers are irrational then. They should create monetary incentives or moral atmospheres that help goalkeepers and kickers to take the best decision for the team. In a way, football teams have a management problem!
Some game theory anybody?
Hugo hints that the optimal strategies depends on the expectation of the opponent. Indeed, most games implies speculating on team members\' and opponents\' expectations. This is because winning strategies require to fulfil team members\' expectations and thwart opponents\' ones.
Suppose that the motivation of kickers and goal-keepers is just to mark and prevent opponents to mark (i.e. forget about the motivation for easy post hoc justification) and suppose (for the sake of the modellisation) that Bar-Ali\'s paper include only advice for the goal keeper so that we have the following strategies:
Goal keepers: stay put or throw yourself on the side.
Kickers: aim at one side or aim at the centre of the gate.
From reading Bar-Ali\'s paper, a goal keeper should infer that, caeteris paribus, one should stay put, even though this is not what they usually do.
Imagine kicker K shoots against goal keeper G. K\'s decision will be dependent of whether
(k0) he (i.e. K) has read Bar-Ali\'s paper [aim at the centre because goal keeper throw themselves on one side most of the time],
(k1) but also whether he (i.e. K) thinks G has read the paper [aim at one side because G having read the paper will stand still] and
(k2) whether he (i.e. K) thinks G think K has read the paper [aim at the centre because G will expect K to aim at one side and will therefore throw himself on one side], and
(k3) whether he (i.e. K) thinks that G thinks that K thinks that G has read the paper [aim at one side because G will stay put since he expects K to aim at the centre because he (i.e. G) thinks that K believes he (i.e. G) will throw himself on one side].
(kn) whether K believes that G assumed that K made n-1 eductive inferences (= inferences about what the other think about what I think) [aim at a side if n is odd, aim at the centre if it is even]
We have the same that goes with G:
(g0) G has read Bar-Ali\'s paper [stay put]
(g1) G has read Bar-Ali\'s paper and thinks that K has read it too [still stay put]
(g2) G has read Bar-Ali\'s paper and think that K has read it too and that he knows that G has read it [throw himself on a side]
(g3) .... [stay put]
(gm) ... [stay put if m is odd and throw himself to the side if m is even]
With all this, G wins when m is odd (stay put) and n is even (aim at the centre) or when m is even and n is odd, which is to say, G wins when m+n is odd.
K wins when m+n is even.
The series does not converge, so common knowledge (when m and n are infinite) is of no help here, which may frustrate some economists.
But I guess it becomes fun only with repeated games with belief updating processes.
Suppose K has a probabilistic belief p about m being odd or even. He updates p each time he can observe the behaviour of G. How does he do that? He can do some bayesian reasoning that confirm that m is odd when he observed G staying put. You can endow G with the same belief updating process.
And at this is the point the game theorist should tell us if he manages to obtain a converging series or not (probably not).
Unfortunately, the bayesian updating procedure could be taken to be quite different: e.g. if G stayed put, then increase the probability that he will throw himself on the side next time, since G will also change its own beliefs about K\'s beliefs ... and all these fun eductive thinking has not been side stepped.
That was some amateur\'s thoughts towards penalty game theory. But for the students of cognition and culture, one should go beyond: the question is how do individuals actually manage to cope with this kind of complexity of our social world: We are pretty good at understanding others\' expectations, but how good? And how many eductive steps means good? And are we good because we do lots of eductive steps or for some other reasons?
Oh! And about the original topic: G and K could always say they thought the other did one more or one less eductive step than they thought. This may be an important characteristics of games to be based on our powerful but bounded social cognitive skills.