Why hyper-inflation? I understand the $MV=PQ$ identity. I also understand the benefit for the government in some cases to having non increasing over time (even if high) values of $\Delta P$ (stealth tax &amp; seigneur-age &amp; real gdp shock,...).

What I do not understand is in what cases would the benefits (for the government) of increasing rates (second derivatives) of inflation out weight the costs?

Because it seems to me many countries (Venezuela, Zimbabwe, apparently Iran right now) could have --yet did not-- hike short term rates (to reduce $\Delta M$ and $\Delta V$) when inflation ($\Delta P$) was going from 'high but stable' to 'out of control' (systematically increasing $\Delta P$).
How central banks raise interest rates

This is a common misunderstanding of central banking.  The popular media portrays things as if the act of increasing interest rates prevents hyperinflation.  In truth, "preventing inflation" and higher interest rates are both side effects of restraining the growth of the money supply.  

What central banks set is not generally interest rates but a target for the interest rate.  In the United States, this is called the federal funds rate target.  Let's break this down.  Funds are simply money loaned from one bank to another, generally on an overnight basis.  The funds rate is the (weighted) average interest rate at which they loan to each other.  It's the federal funds rate because it is at the national level, not limited to one state.  That number is calculated based on market actions.  It is not government controlled.  

The Federal Reserve sets a target for that statistic.  If the statistic is below the target, the Fed sells bonds.  Too high, the Fed buys treasury bonds.  That puts more money in the system.  With more money, more banks have excess reserves (to loan to other banks) and fewer banks are short on reserves (needing to borrow from other banks).  As a result, the average rate falls.  Inflation increases.  Too low, and the opposite happens:  sell bonds; less money; less excess reserves; more borrowers; higher interest rates; lower inflation.  

Alternative tools of monetary policy

Are there other tools that a central bank can use?  Sure.  A central bank can change the reserve fraction, pay interest on excess reserves, or change the rate at which it loans to banks.  

In the US, the rate at which the Fed loans to banks is called the discount rate.  It is generally set to around the same value as the federal funds rate target.  Banks hardly ever borrow from the Fed, as the rules around it make it undesirable.  They would generally prefer to pay a premium and borrow from other banks.  But there is a limit to the premium that they will pay.  

Changing the reserve fraction means changing the amount of deposits that banks must hold.  The higher it is, the less that banks can lend.  This is hard to change suddenly.  If the reserve fraction was 10% (as it currently is in the US) and is raised to 11%, that means that most every bank will have to borrow and few will have money to lend.  This would force many to borrow money from the Fed or be forced to suddenly sell assets.  They would prefer to have some notice.  Then they can save up to have 11% by the announced date.  

Paying interest on excess reserves has a similar effect to changing the reserve fraction, as it increases the amount of money held as reserves.  It is more gradual than changing the reserve fraction.  However, it can also compete with banks that are trying to borrow from other banks.  The other banks may prefer the deposit interest to the loan interest.  Paying interest also costs money.  

Monetary policy and hyperinflation

Let's get back to hyperinflation countries.  The governments of those countries don't want to pay interest on excess reserves.  Even if they didn't mind the other side effects, those countries don't have the money to spend.  And they generally aren't constrained by the reserve fraction.  In a hyperinflation, times are uncertain.  Banks don't want to get stuck loaning money at an interest rate that will be lower than tomorrow's market rate.  But rates are always increasing.  So banks often refuse to loan even if they are awash in reserves.  

For the same reason, central banks don't increase the reserve fraction.  It won't help.  And in a hyperinflation, the central bank would want to increase the discount rate and loan less.  But as already said, central bank to bank loans are rare.  Even eliminating them will not have much of an effect.  

We're back to the one tool remaining, open market operations to try to meet a higher interest rate target.  But for those to work, the central bank has to sell government bonds in competition with the government.  The central bank has a limited amount of assets that it can sell.  Once it is out, it's helpless.  It can set any interest rate target it wants, but it lacks any way to hit it.  

Meanwhile, the government retains plenty of tools to cause inflation.  For example, they can print money.  And they are likely to do so, as everything is becoming more expensive.  So they need more money just to offer the same level of services as they did previously.  So they print it.  But then everything is more expensive and they need more money.  

Effective policy

The solution of course is to lower the level of government spending instead of trying to maintain it against inflation.  But in the short term, that is extremely painful for all those that suddenly don't get the level of government support to which they are accustomed.  

They still need to do that eventually.  Because in a hyperinflation, everything only gets worse.  Because they are trying to trade a permanent increase in inflation for a temporary improvement in the economy.  Obviously trading a permanent harm for a temporary benefit is a bad deal.  But in the short term, it's hard to make that decision and go for the permanent benefit in exchange for the temporary harm.  And the longer this goes, the larger the short term harm is.  And desperate people think more about the immediate term than the long term.  That's part of what it means to be desperate.  

TL;DR:  in the short term, benefits of more money seem to outweigh the harm of inflation.  And in a hyperinflation, people tend to think in the short term, trying to survive today while letting tomorrow fend for itself.